Angles in standard position In the coordinate plane, an angle is in standard position if its vertex is located at the origin, and its initial side lies on the positive x-axis.

Angles are measured by radian or degree, and the value of an angle can be any real number. Positive angles are measured when terminal side rotates counterclockwise from initial side, and negative angles measured in clockwise direction.

An angle of a full rotation is 360° or \(2π\) radians, and an angle of \(n\) full rotations is \(360n\)° or \(2nπ\) radians for an integer \(n=0,±1,±2,\cdots\).

Radians and degrees Given a central angle \(\theta\) (measured by radian) in a circle of radius \(r\) (in standard position), the length \(l\) of the corresponding arc is \(l=r\theta⇒\theta=\frac{l}{r}\) radian. In a unit circle, \(r=1, l=\theta\) radians, which implies the arc length is equivalent to its central angle in radians.

Moreover, its circumference of \(2π\) radians corresponds to a central angle 360°. By this relation, we have the following formulas.
(1) Convert radian to degree \(1 \text{ rad}=\frac{180°}{π}≈57.296° ⇒x \text{ rad} = x\frac{180°}{π}\).
(2) Convert degree to radian \(1° = \frac{π}{180}≈0.017 \text{ rad} ⇒ x° = x\frac{π}{180}\text{ rad}\).
These two formulas are used for conversion between radian and degree.

Examples
(1) Convert degrees to radians \(0°=0;30°=\frac{π}{6};45°=\frac{π}{4};60°=\frac{π}{3};90°= \frac{π}{2};120°=\frac{2π}{3};135°=\frac{3π}{4}\); \(150°=\frac{5π}{6};180°=π;210°=\frac{7π}{6};225°= \frac{5π}{4};240°=\frac{4π}{3};270°=\frac{3π}{2};300°=\frac{5π}{3};315°=\frac{7π}{4};\)\(330°=\frac{11π}{6}; 720°=4π;540°=3π\); \(900°=5π;1080°=6π\).
(2) 0*pi/180 || 30*pi/180 || 45*pi/180 || 60*pi/180 || 90*pi/180 || 120*pi/180 || 135*pi/180 || 150*pi/180 || 180*pi/180 || 210*pi/180 || 225*pi/180 || 240*pi/180 || 270*pi/180 || 300*pi/180 || 315*pi/180 || 330*pi/180 || 720*pi/180 || 540*pi/180 || 900*pi/180 || 1080*pi/180 ||.
(3) Convert radians to degrees. \(\frac{π}{5}=36°;\frac{3π}{5}=108°;\frac{3π}{8}=67.5°;\frac{π}{15}=12°;\frac{7π}{15}=84°;\frac{π}{9}=20°;\frac{5π}{9}=100°\); \(\frac{5π}{18}=50°;\frac{7π}{9}=140°;\frac{5π}{36}=25°\).
(4) pi/5*180/pi || 3*pi/5*180/pi || 3*pi/8*180/pi || pi/15*180/pi || 7*pi/15*180/pi || pi/9*180/pi || 5*pi/9*180/pi || 5*pi/18*180/pi || 7*pi/9*180/pi || 5*pi/36*180/pi || pi/10*180/pi || 9*pi/10*180/pi || pi/12*180/pi || 3*pi/20*180/pi ||.
(5) Convert 27.63° = 27.63·\(\frac{π}{180}\) = 0.1535\(π\) ≈ 0.48 rad, and 1.19 rad ≈ 1.19·\(\frac{180}{π}=(\frac{214.2}{π})\)° ≈ 68.18°. Check 27.63*pi/180 || 1.19*180/pi || 27.63*3.141693/180 || 1.19*180/3.141593 ||.

Negative angles The conversion between degree and radian is the same for negative angles.

Examples
(1) \(-30°=-30·\frac{π}{180}=-\frac{π}{6};-50°=-\frac{5π}{18};-320°=-\frac{8π}{9};-36°= -\frac{π}{5}\). Similarly, \(\frac{-π}{8}=\frac{-π}{8}·\frac{180}{π}=-22.5°\); \(\frac{-5π}{12}=-75°;\frac{-7π}{30}=-42°; \frac{-π}{24}=-7.5°;\frac{-4π}{15}=-48°\).
(2) -30*pi/180 || -50*pi/180 || -320*pi/180 || -36*pi/180 || -pi/8*180/pi || -5*pi/12*180/pi || -7*pi/30*180/pi || -pi/24*180/pi || -4*pi/15*180/pi ||.

Coterminal angles If two angles in standard position have the same initial and terminal sides, they are called coterminal angles. If two angles with opposite signs are coterminal, the sum of their magnitudes is 360° or 2π.

Examples
(1) angles of -50° and 310° have a common terminal side. These pairs of angles share common terminal sides: -20° and 340°, -325° and 35°, -180° and 180°, 90° and -270°, -160° and 200°, -237° and 123°.
(2) If measured by radians, angles of -1.5\(π\) and 0.5\(π\) share common terminal sides, and the pairs of angles -1.72\(π\) and 0.28\(π\), -0.8\(π\) and 1.2\(π\), -1.6\(π\) and 0.4\(π\), -\(π\) and \(π\) and so on have common terminal sides.
(3) Two positive angles are coterminal if they differ by 360° or a multiple of 360°. Thus, 40° and 400° are coterminal, and 70° and 430° are coterminal; \(π\) and \(3π\) coterminal, \(\frac{π}{3}\) and \(\frac{7π}{3}\) are coterminal.

Reference angles The reference angle is the acute angle (the smallest angle) formed by the terminal side of an angle \(0≤θ≤2π\) and the x-axis. Reference angles may appear in all four quadrants.

Angles in quadrant I are their own reference angles. The reference angle of \(θ\) in quadrant II is \(π-θ\), in quadrant III is \(θ-π\), and in quadrant IV is \(2π-θ\). If \(θ≥2π\), find an equivalent angle between 0 and \(2π\) by adding or subtracting \(2nπ\) for \(n\) an integer.

Examples The reference angle of 120° is 60°, the reference angle of 225° is 45°, and the reference angle of 330° is 30°. In radian, the reference angle of \(\frac{3π}{4}\) is \(\frac{π}{4}\), the reference angle of \(\frac{4π}{3}\) is \(\frac{π}{3}\), and the reference angle of \(\frac{7π}{4}\) is \(\frac{π}{4}\).

Graph angles in standard position Use "pln" module to graph an angle in standard position.

Examples The vertex of an angle \(θ\) (measured in radians) in standard position is (0, 0), the initial side can be the line segment joining (0, 0) and (10, 0), and the terminal side can be the line segment joining (0, 0) and \((10\cosθ,10\sinθ)\). Let \(θ=\frac{π}{3}\). Then \(10\cos\frac{π}{3}=5,10\sin\frac{π}{3}≈8.66\), so pln;ln=[(10,0),(0,0),(5,8.66)] || displays the angle of 60° in standard position.

Examples
(1) To graph coterminal angles \(-\frac{\pi}{6}\) and \(\frac{11π}{6}\), calculate \(10\cos(-\frac{\pi}{6})=10\cos\frac{11π}{6}≈8.66,10\sin(-\frac{\pi}{6})\)
\(=10\sin\frac{11π}{6}=-5\). Check pln;ln=[(10,0),(0,0),(8.66,-5)] ||.
(2) Check other examples pln;ln=[(10,0),(0,0),(0,10)] || pln;ln=[(10,0),(0,0),(-10,0)] || pln;ln=[(10,0),(0,0),(0,-10)] || pln;ln=[(10,0),(0,0),(7.071,7.071)] ||.

Trigonometric functions

Definition Given a right triangle, let the leg on the x-axis be \(a\), the other leg be \(b\), and the hypotenuse be \(c\). Denote the acute angle in standard position as \(θ\). Then the six trigonometric functions are defined as follows. \(\sinθ=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{b}{c},\cosθ=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{a}{c},\tanθ=\frac{\sinθ}{\cosθ},\cotθ=\frac{\cosθ}{\sinθ},\secθ=\frac{1}{\cosθ},\cscθ=\frac{1}{\sinθ}\).

Examples
(1) If \(a=8,b=6\), then \(c=10\). Refer to a right triangle by pln;pg=[(0,0),(8,0),(8,6)] ||. By definition, \(\sinθ=\frac{6}{10},\cosθ=\frac{8}{10},\tanθ=\frac{6}{8},\cotθ=\frac{8}{6},\cscθ=\frac{10}{6},\secθ=\frac{10}{8}\).
(2) If the other acute angle in the right triangle is denoted as \(α\), then \(\sinα=\frac{8}{10},\cosα=\frac{6}{10}, \tanα=\frac{8}{6}\),\(\cotα=\frac{6}{8},\cscα=\frac{10}{8},\secα=\frac{10}{6}\) by definition.

Trigonometric functions relate acute angles to the ratios of sides in right triangles. This definition only involve positive and acute angles.

To extend trigonometric functions to all real values, let \((x,y)\) be any point on a circle of radius \(r>0\) in standard position. Then the circle has an equation \(x^2+y^2=r^2\). Suppose \(θ\) is a central angle formed by the positive x-axis, and the terminal side from vertex (0, 0) to the point \((x,y)\) on the circle. Then the six trigonometric functions are \(\sinθ=\frac{y}{r}\), \(\cosθ=\frac{x}{r},\tanθ=\frac{y}{x},\cotθ=\frac{x}{y},\cscθ=\frac{r}{y},\secθ=\frac{r}{x}\).

In particular, if the circle is a unit circle, \(r=1,x^2+y^2=1\), and the six trigonometric functions are as follows: \(\sinθ=y,\cosθ=x,\tanθ=\frac{y}{x},\cotθ=\frac{x}{y},\cscθ=\frac{1}{y},\secθ=\frac{1}{x}\).

Evaluation by definition Each of the six functions can be determined by the coordinates \((x,y)\) of a point on a circle in standard position.

Examples
(1) If \((x,y)\) = (1, 0), then \(\sinθ=0,\cosθ=1,\tanθ=0,\secθ=1\), and both \(\cotθ\) and \(\cscθ\) are undefined.
(2) If \((x,y)\) = (0, 1), then \(\sinθ=1,\cosθ=0,\cotθ=0,\cscθ=1\), and both \(\tanθ\) and \(\secθ\) are undefined.
(3) If \((x,y)\) = (-1, 0), then \(\sinθ=0,\cosθ=-1,\tanθ=0,\secθ=-1\), and both \(\cotθ\) and \(\cscθ\) are undefined.
(4) If \((x,y)\) = (0, -1), then \(\sinθ=-1,\cosθ=0,\cotθ=0, \cscθ=-1\), and both \(\tanθ\) and \(\secθ\) are undefined.
(5) \((x,y)\) = (5, 12), \(⇒r=13, \sinθ=\frac{12}{13}, \cosθ=\frac{5}{13},\tanθ=\frac{12}{5},\cotθ=\frac{5}{12},\cscθ=\frac{13}{12},\secθ=\frac{13}{5}\).
(6) \((x,y)= (-5, 8)⇒r=\sqrt{89}, \sinθ=\frac{8}{\sqrt{89}},\cosθ=\frac{-5}{\sqrt{89}},\tanθ=\frac{8}{-5},\cotθ=\frac{-5}{8}\), \(\cscθ=\frac{\sqrt{89}}{8},\secθ=\frac{\sqrt{89}}{-5}\).
(7) \((x,y)=(\frac{-1}{2},\frac{-\sqrt{3}}{2})⇒r=1, \sinθ= \frac{-\sqrt{3}}{2},\cosθ=\frac{-1}{2},\tanθ=\sqrt{3},\cotθ=\frac{\sqrt{3}}{3}\), \(\cscθ=\frac{-2}{\sqrt{3}},\secθ=-2\).
(8) \((x,y)=(7, -4)⇒r=\sqrt{65}, \sinθ=\frac{-4}{\sqrt{65}},\cosθ=\frac{7}{\sqrt{65}},\tanθ=\frac{-4}{7},\cotθ= \frac{7}{-4}\), \(\cscθ=\frac{\sqrt{65}}{-4},\secθ=\frac{\sqrt{65}}{7}\).

Trigonometric functions for some special angles Most values of trigonometric functions are irrational. But there are a small portion of trigonometric functions having exact values for some special angles.

Examples Check sin(0) || sin(pi/6) || sin(pi/4) || sin(pi/3) || sin(pi/2) || sin(2*pi/3) || sin(3*pi/4) || sin(5*pi/6) || sin(pi) || sin(7*pi/6) || sin(4*pi/3) || sin(5*pi/4) || sin(3*pi/2) || sin(11*pi/6) || sin(5*pi/3) || sin(7*pi/4) || sin(2*pi) || cos(0) || cos(pi/6) || cos(pi/4) || cos(pi/3) || cos(pi/2) || cos(2*pi/3) || cos(3*pi/4) || cos(5*pi/6) || cos(pi) || cos(7*pi/6) || cos(4*pi/3) || cos(5*pi/4) || cos(3*pi/2) || cos(11*pi/6) || cos(5*pi/3) || cos(7*pi/4) || cos(2*pi) || tan(0) || tan(pi/6) || tan(pi/4) || tan(pi/3) || tan(pi/2) || tan(2*pi/3) || tan(3*pi/4) || tan(5*pi/6) || tan(pi) || tan(7*pi/6) || tan(4*pi/3) || tan(5*pi/4) || tan(3*pi/2) || tan(11*pi/6) || tan(5*pi/3) || tan(7*pi/4) || tan(2*pi) || cot(0) || cot(pi/6) || cot(pi/4) || cot(pi/3) || cot(pi/2) || cot(2*pi/3) || cot(3*pi/4) || cot(5*pi/6) || cot(pi) || cot(7*pi/6) || cot(4*pi/3) || cot(5*pi/4) || cot(3*pi/2) || cot(11*pi/6) || cot(5*pi/3) || cot(7*pi/4) || cot(2*pi) || csc(0) || csc(pi/6) || csc(pi/4) || csc(pi/3) || csc(pi/2) || csc(2*pi/3) || csc(3*pi/4) || csc(5*pi/6) || csc(pi) || csc(7*pi/6) || csc(4*pi/3) || csc(5*pi/4) || csc(3*pi/2) || csc(11*pi/6) || csc(5*pi/3) || csc(7*pi/4) || csc(2*pi) || sec(0) || sec(pi/6) || sec(pi/4) || sec(pi/3) || sec(pi/2) || sec(2*pi/3) || sec(3*pi/4) || sec(5*pi/6) || sec(pi) || sec(7*pi/6) || sec(4*pi/3) || sec(5*pi/4) || sec(3*pi/2) || sec(11*pi/6) || sec(5*pi/3) || sec(7*pi/4) || sec(2*pi) ||.

Examples Check other values and relations such as sin(2*pi/5) || sin(1) || sin(1.0) || cos(0.43) || tan(3.4) || cot(1.8) || sec(0.36) || csc(-2.79) || sin(2)>cos(2) || sin(0.6)>cos(0.6) || tan(4.5)>cot(4.5) || tan(3)>tan(4) || sin(2)>sin(2.5) ||.

Domain and range The functions \(\sin x\) and \(\cos x\) are defined for all \(x∈(-∞, ∞)\) and have a range [-1, 1].

The function \(\tan x\) is defined on \(x∈(-\frac{π}{2}+kπ, \frac{π}{2}+kπ)\), and \(\cot x\) on \(x∈(kπ, π+kπ)\), where \(k\) is an integer. The range of \(\tan x\) or \(\cot x\) is \((-∞,∞)\). The function \(\sec x\) is defined for all real numbers except \(x=\frac{π}{2}+kπ\), and \(\csc x\) is defined for all \(x\) except \(x=kπ\), where \(k\) is an integer. The range of \(\sec x\) and \(\csc x\) is \((-∞,-1] ∪[1,∞)\).

Keep in mind that the domain of \(\tan x\) or \(\sec x\) is \(\{x|x≠ \frac{π}{2}+kπ\}\) or \(x\) is not in the set \(\{\pm \frac{π}{2},\pm \frac{3π}{2}, \pm\frac{5π}{2}, \cdots\}\), and the domain of \(\cot x\) or \(\csc x\) is \(\{x| x≠ kπ\}\) or \(x\) is not in the set \(\{0, \pm π, \pm 2π, 3π, 4π,\cdots\}\).

Even and odd Among the six trigonometric functions, cosine and secant functions are even and the rest are odd functions. Thus, \(\cos(-x)=\cos x, \sec(-x)=\sec x,\sin(-x)=-\sin x,\tan(-x)=-\tan x\),
\(\cot(-x)=-\cot x,\csc(-x)=-\csc x\). So \(\cos x\) and \(\sec x\) are symmetric about the y-axis and the rest are symmetric about the origin.

Examples Check plt;cos(x) || plt;sec(x) || plt;sin(x) || plt;tan(x) || plt;cot(x) || plt;csc(x) || sin(-2) || cos(-3) || tan(-4) || cot(-5) || sec(-7) || csc(-8) ||.

Signs of trigonometric functions The sign of trigonometric function is determined by the terminal side of a given angle (in standard position). Among the six functions, each pair of reciprocal functions has same sign no matter which quadrant the angle lies in. That is, \(\sin x\) and \(\csc x\) always have same sign, \(\cos x\) and \(\sec x\) always have same sign, and \(\tan x\) and \(\cot x\) always have same sign.

The acronym "ASTC" summarizes the positive "+" sign a trigonometric function appears in the four quadrants:
quadrant I (all);  quadrant II (\(\sin x,\csc x\));  quadrant III (\(\tan x,\cot x\));  quadrant IV (\(\cos x,\sec x\)).
If a function does not have "+" sign in a quadrant, it must have "-" sign in that quadrant. All the six functions have "+" sign in the first quadrant, and only one pair of reciprocal functions has "+" in each of the rest quadrants. These rules apply to negative angles.

Angle terminal side lies in a quadrant Let \(0≤x≤2π\) be an angle in standard position.

If the terminal side of \(x\) is in quadrant I, then \(0< x <\frac{π}{2}\) and all six functions are positive.

If the terminal side of \(x\) lies in quadrant II, then \(\frac{π}{2} < x < π,\sin x>0,\csc x>0\), and the rest four functions are negative, or \(\cos x< 0, \tan x < 0, \cot x < 0, \sec x < 0\).

If the terminal side of \(x\) lies on quadrant III, \(π< x < \frac{3π}{2},\tan x>0,\cot x>0\), and the rest four are negative, or \(\sin x< 0, \cos x< 0,\sec x< 0\), \(\csc x< 0\).

If the terminal side of \(x\) lies in quadrant IV, then \(\frac{3π}{2}< x< 2π,\cos x> 0,\sec x>0\), and the rest are negative, or \(\sin x < 0,\csc x< 0,\tan x< 0, \cot x< 0\).

Angle terminal side lies on coordinate axes Suppose the terminal side of \(x\) lies on the coordinate axes.

If it lies on the positive x-axis, \(x=0,\sin0=0\), \(\cos0=1,\tan0=0,\sec0=1\) and \(\cot0,\csc0\) are undefined.

If it lies on the positive y-axis, \(x=\frac{π}{2},\sin\frac{π}{2}=1\), \(\cos\frac{π}{2}=0,\csc\frac{π}{2}=1,\cot\frac{π}{2}=0\) and \(\tan\frac{π}{2},\sec\frac{π}{2}\) are undefined.

If it lies on the negative x-axis, \(x=π,\sinπ=0\), \(\cosπ=-1,\tanπ=0,\secπ=-1\) and \(\cotπ,\cscπ\) are undefined.

If it lies on the negative y-axis, \(x=\frac{3π}{2}\), \(\sin\frac{3π}{2}=-1,\cos\frac{3π}{2}=0,\csc\frac{3π}{2}=-1,\cot\frac{3π}{2}=0\) and \(\tan\frac{3π}{2},\sec\frac{3π}{2}\) are undefined.

Examples
(1) Since \(x=\frac{π}{6}\) is in quadrant I, all six functions are positive. Check sin(pi/6) || cos(pi/6) || tan(pi/6) || cot(pi/6) || sec(pi/6) || csc(pi/6) ||.
(2) Since \(x=\frac{2π}{3}\) is in quadrant II, only \(\sin x\) and \(\csc x\) are positive. Check sin(2*pi/3) || cos(2*pi/3) || tan(2*pi/3) || cot(2*pi/3) || sec(2*pi/3) || csc(2*pi/3) ||.
(3) Since \(x=\frac{5π}{4}\) is in quadrant II, only \(\tan x\) and \(\cot x\) are positive, check sin(5*pi/4) || cos(5*pi/4) || tan(5*pi/4) || cot(5*pi/4) || sec(5*pi/4) || csc(5*pi/4) ||.
(4)Since \(x=\frac{11π}{6}\) is in quadrant IV, only \(\cos x\) and \(\sec x\) are positive. Check sin(11*pi/6) || cos(11*pi/6) || tan(11*pi/6) || cot(11*pi/6) || sec(11*pi/6) || csc(11*pi/6) ||.

Let \(-2π≤x≤0\) be a negative angle in standard position. Since \(\cos x\) and \(\sec x\) are even functions, \(\cos(-x)=\cos x\) and \(\sec(-x)=\sec x\), and the rest four function have \(\sin(-x)=-\sin x,\csc(-x)=-\csc x\),
\(\tan(-x)=-\tan x,\cot(-x)=-\cot x\). We can also use these facts to determine the sign of trigonometric functions with negative angles.

Examples
(1) \(\cos\frac{-π}{3}=\cos\frac{π}{3}=\frac{1}{2}\) because \(\frac{-π}{3}\) is in quadrant IV and \(\frac{π}{3}\) in quadrant I and \(\cos x\) has positive sign in both quadrants. Check cos(-pi/3) ||.
(2) \(\tan(\frac{-3π}{4})=-\tan(\frac{3π}{4})=-(-1)=1\) because \(\frac{-3π}{4}\) is in quadrant III. Check tan(-3*pi/4) ||.
(3) \(\sin(\frac{-5π}{3})=-\sin(\frac{5π}{3})=-(-\frac{\sqrt{3}}{2})=\frac{\sqrt{3}}{2}\) because \(\frac{-5π}{3}\) is in quadrant II. Check sin(-5*pi/3) ||.
(4) \(\csc(\frac{-7π}{4})=-\csc(\frac{7π}{4})=-(-2)=\sqrt{2}\) because \(\frac{-7π}{4}\) is in quadrant I. Check csc(-7*pi/4) ||.

Evaluating trigonometric functions We can use the reference angle (always acute) to evaluate trigonometric functions for any angle by the following steps,
(i) find the corresponding reference angle,
(ii) evaluate the function with the refernce angle, and
(iii) determine the sign of the value.

Examples
(1) \(\sin\frac{3π}{4}=\sin\frac{π}{4}=\frac{\sqrt{2}}{2}\) because the reference angle of \(\frac{3π}{4}\) is \(\frac{π}{4}\), and \(\frac{3π}{4}\) is in quadrant II where the sign of \(\sin x\) is "+". Check sin(3*pi/4) || sin(3*pi/4)==sin(pi/4) ||. \(\cos\frac{2π}{3}=-\cos\frac{π}{3}=\frac{-1}{2}\) because the reference angle of \(\frac{2π}{3}\) is \(\frac{π}{3}\), and \(\frac{2π}{3}\) is in quadrant II where the sign of \(\cos x\) is "-". Check cos(2*pi/3) || cos(2*pi/3)==-cos(pi/3) ||.

(2) \(\tan\frac{7π}{6}=\tan\frac{π}{6}=\frac{\sqrt{3}}{3}\) because the reference angle of \(\frac{7π}{6}\) is \(\frac{π}{6}\), and \(\frac{7π}{6}\) is in quadrant III where the sign of \(\tan x\) is "+". Check tan(7*pi/6) || tan(7*pi/6)==tan(pi/6) ||. \(\sin\frac{4π}{3}=-\sin\frac{π}{3}=-\frac{\sqrt{3}}{2}\). The reference angle of \(\frac{4π}{3}\) is \(\frac{π}{3}\), and \(\frac{4π}{3}\) is in quadrant III where the sign of \(\sin x\) is "-". Check sin(4*pi/3) || sin(4*pi/3)==-sin(pi/3) ||.

(3) \(\sec\frac{11π}{6}=\sec\frac{π}{6}=\frac{2\sqrt{3}}{3}\). The reference angle of \(\frac{11π}{6}\) is \(\frac{π}{6}\), and \(\frac{11π}{6}\) is in quadrant IV where the sign of \(\sec x\) is "+". Check sec(11*pi/6) || sec(11*pi/6)==sec(pi/6) ||. \(\cot\frac{7π}{4}=-\cot\frac{π}{4}=-1\). The reference angle of \(\frac{7π}{4}\) is \(\frac{π}{4}\), and \(\frac{7π}{4}\) is in quadrant IV where the sign of \(\cot x\) is "-". Check cot(7*pi/4) || cot(7*pi/4)==-cot(pi/4) ||.

Examples If an angle \(-2π≤x≤0\) is negative, first apply the property of even or odd function, and then evaluate the function with an angle within 0 and \(2π\).
(1) \(\sin(-\frac{5π}{6})=-\sin\frac{5π}{6}=-(-\sin\frac{π}{6})=\frac{1}{2}\). Check sin(-5*pi/6) || sin(-5*pi/6)==sin(pi/6) ||.
(2) \(\cos(-\frac{5π}{3})=\cos\frac{5π}{3}=\cos\frac{π}{3}=\frac{1}{2}\). Check cos(-5*pi/3) || cos(-5*pi/3)==cos(pi/3) ||.
(3) \(\tan(-\frac{3π}{4})=-\tan\frac{3π}{4}=-(-\tan\frac{π}{4})=1\). Check tan(-3*pi/4) || tan(-3*pi/4)==tan(pi/4) ||.

Examples If an angle \(x\) is out of the interval \([-2π,2π]\), first find a corresponding coterminal angle of \(x\) by adding or subtracting \(2nπ\) for \(n\) integer.
(1) \(\sin\frac{7π}{3}=\sin(\frac{π}{3}+2π)=\sin\frac{π}{3}=\frac{\sqrt{3}}{2}\) since \(\frac{7π}{3}\) and \(\frac{π}{3}\) are coterminal. || sin(7*pi/3) || sin(7*pi/3)==sin(pi/3) ||.
(2) \(\cos\frac{17π}{6}=\cos(\frac{5π}{6}+2π)=\cos\frac{5π}{6}=-\cos\frac{π}{6}=-\frac{\sqrt{3}}{2}\) because \(\frac{17π}{6}\) and \(\frac{5π}{6}\) are coterminal, \(\frac{π}{6}\) is the reference angle of \(\frac{5π}{6}\), and \(\cos\frac{5π}{6}\) and \(\cos\frac{π}{6}\) differ only by a sign. Check cos(17*pi/6) || cos(17*pi/6)==-cos(pi/6) ||.
(3) \(\sin\frac{11π}{5}=\sin(\frac{11π}{5}-2π)\)\(=\sin\frac{π}{5}\). Check sin(11*pi/5)==sin(pi/5) ||.

Examples sin(8*pi/7)==sin(pi/7) || tan(9*pi/5)==-tan(pi/5) || cos(13*pi/6)==cos(pi/6) || sin(7*pi/3)==-sin(pi/3) || tan(15*pi/4)==-tan(pi/4) || cos(17*pi/9)==cos(pi/9) || sin(15*pi/8)==sin(pi/8) || tan(13*pi/7)==tan(pi/7) || cot(9*pi/10)==cot(pi/10) || sec(20*pi/11)==sec(2*pi/11) || csc(31*pi/15)==csc(pi/11) || cot(27*pi/14)==-cot(pi/14) ||.

Graphs of trigonometric functions The graph of \(y=\sin(-x)=-\sin x\) can be obtained by reflecting the graph of \(\sin x\) across either the y-axis or x-axis. Similarly, we can do other transformations (shifting and scaling) to all trigonometric functions.

Examples
(1) Check plt;sin(x);sin(-x) || plt;sin(x);-sin(x) ||. This is also true for \(\tan x,\cot x,\csc x\). Check plt;tan(x);tan(-x);itv=(-1.56,1.56) || plt;tan(x);-tan(x);itv=(-1.56,1.56) || plt;cot(x);cot(-x);-cot(x);itv=(0,3.14) || plt;csc(x);csc(-x);-csc(x);itv=(0.05,3.1) ||.
(2) The graph of \(y=-\cos x\) can be obtained by reflecting the graph of \(\cos x\) across the x-axis. Check plt;cos(x);-cos(x) ||. This is also true for \(\sec x\). Check plt;sec(x);-sec(x) ||.
(3) Check other graphs involves trigonometric functions plt;x+sin(x) || plt;cos(x)-x || plt;5*sin(10/x) || plt;x*sin(x) || plt;x*cos(x) || plt;sin(x)/x || plt;(1-cos(x))/x || plt;x+tan(x) || plt;x*tan(x) || plt;tan(x)/x || plt;abs(sin(x)) || plt;1+abs(cos(x)) ||.

Examples The graph of \(y=4\sin(3x-2)-1\) can be obtained by first horizontally compressing the graph of \(\sin x\) by a factor of 3, shifting it to the right 2/3 units, then vertically stretching it by a factor of 2, and finally shifting it down 1 unit. Check plt;sin(x);4*sin(3*x-2)-1 ||.

Examples The graph of \(y=-3\cos(\frac{x+1}{2})+2\) can be obtained by first horizontally stretching the graph of \(\cos x\) by a factor of 2, shifting it to the left 1 unit, then vertically stretching it by a factor of 3, reflecting it across the x-axis, and finally shifting it up 2 units. || plt;cos(x);-3*cos(x/2+1/2)+2 ||.

Examples Similar transformations to other trigonometric functions by plt;tan(-2*x) || plt;cot(3*x/2) || plt;2*csc(x) || plt;sec(0.5*x) || plt;-4*sin(2*x/3) || plt;0.5*tan(x) || plt;2*cot(-x/2) || plt;-2*sec(3*x) ||.

Examples Use graphs to show some inequalities for trigonometric functions.
(1) Refer to the graph plt;sin(x);abs(x) ||, which shows the graph of \(|x|\) is entirely above the graph of \(\sin x\). The graph of \(-|x|\) lies entirely below the graph of \(\sin x\). Check plt;sin(x);-abs(x) ||. In general for any angle \(x\) radians, the inequalities \(|\sin x|≤|x|⇒-|x|≤\sin x ≤ |x|\). If \(x≥0, \sin x < x\), and if \(x< 0, \sin(x)> x\). Check plt;sin(x);x ||.
(2) The inequality \(-|x|≤1-\cos x≤|x|\) hold. Check plt;-abs(x);1-cos(x) || plt;1-cos(x);abs(x) ||.
(3) If \(0< x < \frac{π}{2},\sin x < x < \tan x\), and if \(-\frac{π}{2}< x< 0,\tan x < x < \sin x\). Check plt;x;tan(x);sin(x);itv=(-1.56,1.56) ||.

Period and amplitude All the six trigonometric functions are periodic. A function \(f\) is periodic if \(f(x+T)=f(x)\) for all \(x\), and the smallest positive number of \(T\) is called the period of \(f\). Graphs of tangent and cotangent show their period is \(T=π\), and the rest have a period \(T=2π\). So \(\sin(x+2π)=\sin x,\cos(x+2π)=\cos x\), \(\sec(x+2π)=\sec x,\csc(x+2π)=\csc x\), and \(\tan(x+π)=\tan x,\cot(x+π)=\cot x\).

Examples Check sin(x+2*pi) || cos(x+2*pi) || tan(x+pi) || cot(x+pi) || sec(x+2*pi) || csc(x+2*pi) ||.

Transforming trigonometric functions Just as algebraic functions, trigonometric functions can be transformed in various ways to form new functions. A sinusoidal function in form \(f(x)=A\sin\bigl (B(x-C)\bigr)+D\) is a transformation of \(\sin x\) or \(\cos x\). The constant \(A\) is called the amplitude of \(f,B\) changes the period \(2π\) of \(\sin x\) to a period \(\frac{2π}{|B|},C\) gives a horizontal shift, and \(D\) causes a vertical shift.

Examples Check sin((x+4*pi)/2) || cos(3*(x+2*pi/3)) || sin(2*(x+3*pi)/3) || tan(2*(x+pi/2)) || cot(3*(x+pi/3)) || tan((x+2*pi)/2) || cot(4*(x+3*pi/4)/3) ||.

Examples
(1) \(f(x)=2\sin(0.8x-1)+3\) has a period \(\frac{5π}{2}\) and amplitude 2. Its graph can be obtained by horizontally compressing the graph of \(\sin x\) by a factor of 5/4, shifting it to the right 5/4 units, then vertically stretching it by a factor of 2, and shifting it up 3 units. Check plt;sin(x);2*sin(0.8*x-1)+3 || 2*sin(4*(x+5*pi/2)/5-1)+3 ||.
(2) The period of \(3\cos2x\) is \(π\) and amplitude is 3. Check plt;cos(x);3*cos(2*x) || 3*cos(2*(x+pi)) ||.
(3) The period of \(-1.8\sin(1.5x+2)\) is \(\frac{4π}{3}\) and amplitude is 1.8. Check plt;sin(x);-1.8*sin(1.5*x+2) || -1.8*sin(1.5*(x+4*pi/3)+2) ||.
(4) The period of \(2\sin(1-3x)\) is \(\frac{2π}{3}\) and amplitude is 2. Check plt;sin(x);2*sin(1-3*x) || 2*sin(1-3*(x+2*pi/3)) ||.
(5) The period of \(\frac{7}{4}\cos(2-4x)\) is \(\frac{π}{2}\) and amplitude is 7/4. Check plt;cos(x);7*cos(2-4*x)/4 || 7*cos(2-4*(x+pi/2))/4 ||.
(6) The period of \(\sin2x+\cos x\) is \(2π\). Check plt;sin(2*x)+cos(x) ||.
(7) The period of \(\sin2x+\cos3x\) is \(2π\). Check plt;sin(2*x)+cos(3*x) || sin(2*(x+2*pi))+cos(3*(x+2*pi)) ||.
(8) The period of \(\tan2x\) is \(\frac{π}{2}\), the period of \(\cot\frac{2x}{3}\) is \(\frac{3π}{2}\), and the period of \(\tan\frac{x}{2}\) is \(2π\). Check plt;tan(2*x) || plt;cot(2*x/3) || plt;tan(x/2) ||.

Trigonometric identities

Fundamental identity \(\sin^2x+\cos^2x=1\). Dividing \(\sin^2x, \cos^2x\) to both sides of the identity, we obtain \(1+\tan^2x=\sec^2x\) and \(1+\cot^2=\csc^2x\).

Examples
(1) sin(x)**2+cos(x)**2 || 1+tan(x)**2 || 1+cot(x)**2 || sin(3)**2+cos(3)**2 || sin(5)**2+cos(5)**2 || 1+tan(4)**2==sec(4)**2 || 1+cot(2.5)**2-csc(2.5)**2 ||.

(2) Since \(\sin^21+\cos^21=1,\sqrt{1-\cos^21}=\sqrt{\sin^21}=\sin1\) because \(\sin1>0\). Check (1-cos(1)**2)**(1/2) ||. Similarly, \(\sqrt{1-\sin^22}=\sqrt{\cos^22}=-\cos2\) because \(\cos 2< 0\). Check (1-sin(2)**2)**(1/2) ||.

(3) \(\sqrt{1+\tan^23}\)\(=\sqrt{\sec^23}=-\sec3=\frac{-1}{\cos3}\) because \(\sec3< 0\). Check (1+tan(3)**2)**(1/2) ||. In a similarly fashion, \(\sqrt{\csc^21.8π-1}=\sqrt{\cot^21.8π}=-\cot(1.8π)\) because \(\cot1.8π< 0\). Check (csc(1.8*pi)**2-1)**(1/2) ||.

Sum of angles Formulas for sum of angles can be used to shift or compute double, supplementary, and complementary angles. The following are the three basic formulas.
sin(x + y) = sin(x)cos(y) + cos(x)sin(y),  cos(x + y) = cos(x)cos(y) - sin(x)sin(y),  \(\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}\).
Replacing y by -y and applying even and odd properties, we get the formulas for sin(x - y), cos(x - y) and tan(x - y).

Examples sin(x+y)==sin(x)*cos(y)+cos(x)*sin(y) || sin(x-y)==sin(x)*cos(-y)+cos(x)*sin(-y) || cos(x+y)==cos(x)*cos(y)-sin(x)*sin(y) || cos(x-y)==cos(x)*cos(y)+sin(x)*sin(y) || tan(x+y)==(tan(x)+tan(y))/(1-tan(x)*tan(y)) || tan(x-y)==(tan(x)-tan(y))/(1+tan(x)*tan(y)) ||.

Double angles \(\sin 2x=2\sin x\cos x, \cos 2x=\cos^2x-\sin^2x, \tan2x=\frac{2\tan x}{1-\tan^2x}\), which can be obtained by replacing \(y=x\) in the formulas for sum of angles.

Examples sin(2*x)==2*sin(x)*cos(x) || cos(2*x)==cos(x)**2-sin(x)**2 || tan(2*x)==2*tan(x)/(1-tan(x)**2) || cos(4*x)==cos(2*x)**2-sin(2*x)**2 || tan(3*x)==2*tan(1.5*x)/(1-tan(1.5*x)**2) ||.

Half-angle \(\cos^2x=\frac{1+\cos 2x}{2},\sin^2x=\frac{1-\cos 2x}{2},\tan^2x=\frac{1-\cos 2x}{1+\cos 2x},\tan\frac{x}{2}=±\sqrt{\frac{1-\cos x}{1+\cos x}}=\frac{1-\cos x}{\sin x}=\frac{\sin x}{1+\cos x}\).

Examples sin(x)**2==(1-cos(2*x))/2 || cos(x)**2==(1+cos(2*x))/2 || sin(x/2)**2==(1-cos(x))/2 || cos(x/2)**2==(1+cos(x))/2 || tan(x/2)==sin(x)/(1+cos(x)) || tan(x/2)==(1-cos(x))/sin(x) || tan(x)**2==(1-cos(2*x))/(1+cos(2*x)) ||.

Complimentary angles \(\sin(\frac{π}{2}-x)=\cos x,\cos(\frac{π}{2}-x)=\sin x, \tan(\frac{π}{2}-x)=\cot x,\cot(\frac{π}{2}-x)=\tan x\), \(\sec(\frac{π}{2}-x)=\csc x, \csc(\frac{π}{2}-x)=\sec x\).

Examples sin(pi/2-x) || cos(pi/2-x) || tan(pi/2-x) || cot(pi/2-x) || sec(pi/2-x) || csc(pi/2-x) || sin(pi/2-3*x) || cos(pi/2-x/2) || tan(pi/2-3*x/2) || cot(pi/2-2*x/5) ||.

Supplementary angles \(\sin(π-x)=\sin x,\cos(π-x)=-\cos x,\tan(π-x)=-\tan x\), \(\cot(π-x)=-\cot x,\sec(π-x)=-\sec x,\csc(π-x)=\csc x\).

Examples sin(pi-x) || cos(pi-x) || tan(pi-x) || cot(pi-x) || sec(pi-x) || csc(pi-x) || sin(pi-2*x) || cos(pi-1.3*x) || tan(pi-0.5*x) || cot(pi-3*x/4) || sec(pi-7*x/5) || csc(pi-x/4) ||.

Shifting \(\sin(x+\frac{π}{2})=\cos x,\cos(x+\frac{π}{2})=-\sin x,\tan(x+\frac{π}{2})=-\cot x,\cot(x+\frac{π}{2})=-\tan x\), \(\csc(x+\frac{π}{2})=\sec x,\sec(x+\frac{π}{2})=-\csc x\).

Examples sin(x+pi/2) || cos(x+pi/2) || tan(x+pi/2) || cot(x+pi/2) || sec(x+pi/2) || csc(x+pi/2) || sin(2*x+pi/2) || cos(x/3+pi/2) || tan(0.7*x+pi/2) || cot(4*x/5+pi/2) || sec(8*x/3+pi/2) || csc(2.9*x+pi/2) ||.

\(\sin(x+\frac{3π}{2})=-\cos x,\cos(x+\frac{3π}{2})=\sin x,\tan(x+\frac{3π}{2})=-\cot x,\cot(x+\frac{3π}{2})=-\tan x\), \(\csc(x+\frac{3π}{2})=-\sec x,\sec(x+\frac{3π}{2})=\csc x\).

Examples sin(x+3*pi/2) || cos(x+3*pi/2) || tan(x+3*pi/2) || cot(x+3*pi/2) || sec(x+3*pi/2) || csc(x+3*pi/2) || sin(2*x+3*pi/2) || cos(x/3+3*pi/2) || tan(0.7*x+3*pi/2) || cot(4*x/5+3*pi/2) || sec(8*x/3+3*pi/2) || csc(2.9*x+3*pi/2) ||.

\(\sin(x+π)=-\sin x,\cos(x+π)=-\cos x,\tan(x+π)=\tan x,\cot(x+π)=\cot x,\csc(x+π)=-\csc x\), \(\sec(x+π)=\sec x\).

Examples sin(x+pi) || cos(x+pi) || tan(x+pi) || cot(x+pi) || sec(x+pi) || csc(x+pi) || sin(5*x+pi) || cos(x/4+pi) || tan(5.3*x+pi) || cot(7*x/12+pi) || sec(x/9+pi) || csc(3*x/2+pi) ||.

Sum to product \(\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\). Replacing \(y\) by \(-y\), we have \(\sin x-\sin y=2\sin\frac{x-y}{2}\cos\frac{x+y}{2}\). \(\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}, \cos x-\cos y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\).

Examples sin(x)+sin(y)==2*sin((x+y)/2)*cos((x-y)/2) || sin(x)-sin(y)==2*sin((x-y)/2)*cos((x+y)/2) || cos(x)+cos(y)==2*cos((x+y)/2)*cos((x-y)/2) || cos(x)-cos(y)==-2*sin((x+y)/2)*sin((x-y)/2) ||.

Product to sum \(\sin x\sin y =\frac{1}{2}[\cos(x-y)-\cos(x+y)], \cos x\cos y =\frac{1}{2}[\cos(x+y)+\cos(x-y)]\), \(\sin x\cos y =\frac{1}{2}[\sin(x+y)+\sin(x-y)]\).

Examples (sin(x+y)+sin(x-y))/2 || (sin(x+y)-sin(x-y))/2 || (cos(x+y)+cos(x-y))/2 || (cos(x+y)-cos(x-y))/2 ||.

Inverse trigonometric functions The six basic trigonometric functions are periodic, they are not one-to-one, and a function value corresponds to infinitely many angles. For example, \(\sin x=0,x=nπ\) for \(n\) any integer. The six basic inverse trigonometytric functions are defined in the following restricted domains.
\(y=\sin^{-1}x⇔\sin y=x\) and  \(x∈ [-1,1], y∈ [-\frac{π}{2},\frac{π}{2}]= [-90°, 90°]\).
\(y=\cos^{-1}x⇔\cos y=x\) and  \(x∈ [-1,1], y∈ [0, π]=[0°, 180°]\).
\(y=\tan^{-1}x⇔\tan y=x\) and  \(x∈ R, y∈ (-\frac{π}{2},\frac{π}{2})= (-90°, 90°)\).
\(y=\cot^{-1}x⇔\cot y=x\) and  \(x∈R, y∈ (0, π)= (0°, 180°)\).
\(y=\sec^{-1}x⇔\sec y=x\) and  \(|x|≥1, y∈ [0, π]\) and \(y≠ \frac{π}{2}\).
\(y=\csc^{-1}x⇔\csc y=x\) and  \(|x|≥1, y∈ [-\frac{π}{2}, \frac{π}{2}]\) and \(y≠0\).

The composites \(\sin(\sin^{-1}x)=x, \cos(\cos^{-1}x)=x, \tan(\tan^{-1}x)=x, \cot(\cot^{-1}x)=x\),\(\sec(\sec^{-1}x)=x,\csc(\csc^{-1}x)=x\) are correct for all \(x\) in their respective domains.

Examples sin(asin(x)) || cos(acos(x)) || tan(atan(x)) || cot(acot(x)) || sec(asec(x)) || csc(acsc(x)) || sin(asin(0.3)) || cos(acos(-0.4)) || tan(atan(-7)) || cot(acot(10)) || sec(asec(3.5)) || csc(acsc(-21)) ||.

Attention \(\sin^{-1}(\sin x)=x,\cos^{-1}(\cos x)=x,\tan^{-1}(\tan x)=x,\cot^{-1}(\cot x)=x,\sec^{-1}(\sec x)=x\), \(\csc^{-1}(\csc x)=x\) are NOT necessarily true, because \(x\) may not be in the restricted domain where the function is defined to have an inverse. The composites are correct only if \(x\) is in the restricted domain.

Examples
(1) \(\cos^{-1}(\cos\frac{3π}{2}) ≠\frac{3π}{2}\) because \(\frac{3π}{2}\) is not in the restricted domain \([0,π]\) for \(\cos{-1}(x)\). Simplify \(\cos^{-1}(\cos\frac{3π}{2})=\cos^{-1}(\cos\frac{π}{2})=\frac{π}{2}\). Check acos(cos(3*pi/2)) ||.
(2) \(\sin^{-1}(\sin(-1))=-1,\cos^{-1}(\cos(2))=2,\tan^{-1}(\tan(-0.9))=-0.9,\cot^{-1}(\cot(1.5))=1.5\), \(\sec^{-1}(\sec(2.5))=2.5,\csc^{-1}(\csc(-1.3))=-1.3\). Check asin(sin(-1)) || acos(cos(2)) || atan(tan(-0.9)) || acot(cot(1.5)) || asec(sec(2.5)) || acsc(csc(-1.3)) ||.
(3) \(\sin^{-1}(\sin\frac{2π}{3})=\sin^{-1}(\sin\frac{π}{3})=\frac{π}{3},\sin^{-1}(\sin(-\frac{7π}{6}))=\sin^{-1}(\sin(\frac{π}{6}))=\frac{π}{6},\cos^{-1}(\cos(-\frac{π}{4}))\)
\(=\cos^{-1}(\cos(\frac{π}{4}))=\frac{π}{4}=\tan^{-1}(\tan(-\frac{2π}{3}))=\tan^{-1}(\tan(\frac{π}{3}))=\frac{π}{3},\cos^{-1}(\cos(-\frac{4π}{3}))=\cos^{-1}(\cos(\frac{2π}{3}))\)
\(=\frac{2π}{3}\). Check asin(sin(2*pi/3)) || asin(sin(-7*pi/6)) || acos(cos(-pi/4)) || atan(tan(-2*pi/3)) || acos(cos(-4*pi/3)) ||.

Examples \(\cot^{-1}(\cot0.2)=0.2=\frac{π}{2}-\tan^{-1}(\cot0.2),\cot^{-1}(\cot2)=2=\frac{π}{2}-\tan^{-1}(\cot2)\), \(\cot^{-1}(\cot(-3))=\frac{π}{2}-\tan^{-1}(\cot(-3))=π-3\). Check acot(cot(0.2)) || acot(cot(0.2))+atan(cot(0.2)) || pi/2-atan(cot(2)) || acot(cot(pi/2)) || pi/2-atan(cot(-3)) ||.

Evaluate inverse trigonometric functions

Examples
(1) \(\sin^{-1}0=0\),\(\sin^{-1}1=\frac{π}{2},\sin^{-1}(-1)=\frac{-π}{2},\sin^{-1}\frac{1}{2}=\frac{π}{6},\sin^{-1}\frac{\sqrt{2}}{2}=\frac{π}{4},\sin^{-1}\frac{\sqrt{3}}{2}=\frac{π}{3}\), \(\sin^{-1}\frac{-1}{2}=\frac{-π}{6},\sin^{-1}\frac{-\sqrt{2}}{2}=\frac{-π}{4},\sin^{-1}\frac{-\sqrt{3}}{2} =\frac{-π}{3}\). Check asin(0) || asin(1) || asin(-1) || asin(1/2) || asin(-1/2) || asin(3**(1/2)/2) || asin(2**(1/2)/2) || asin(-3**(1/2)/2) || asin(-2**(1/2)/2) ||.

(2) \(\cos^{-1}0=\frac{π}{2}\),\(\cos^{-1}1=0,\cos^{-1}(-1)=π,\cos^{-1}\frac{1}{2}=\frac{π}{3},\cos^{-1}\frac{\sqrt{2}}{2}=\frac{π}{4},\cos^{-1}\frac{\sqrt{3}}{2}=\frac{π}{6},\cos^{-1}\frac{-1}{2}=\frac{2π}{3}\), \(\cos^{-1}\frac{-\sqrt{2}}{2}=\frac{3π}{4},\cos^{-1}\frac{-\sqrt{3}}{2} =\frac{5π}{6}\). Check acos(0) || acos(1) || acos(-1) || acos(1/2) || acos(-1/2) || acos(3**(1/2)/2) || acos(2**(1/2)/2) || acos(-3**(1/2)/2) || acos(-2**(1/2)/2) ||.

(3) \(\tan^{-1}0=0,\tan^{-1}1=\frac{π}{4}\),\(\tan^{-1}(-1)=\frac{-π}{4},\tan^{-1}\frac{\sqrt{3}}{3}=\frac{π}{6},\tan^{-1}\frac{-\sqrt{3}}{3}=\frac{-π}{6},\tan^{-1}\sqrt{3}=\frac{π}{3}\), \(\tan^{-1}(-\sqrt{3})=\frac{-π}{3}\). Check atan(0) || atan(1) || atan(-1) || atan(3**(1/2)/3) || atan(3**(1/2)) || atan(-3**(1/2)/3) || atan(-3**(1/2)) ||.

Attention Note that the "acot" function here (or \(\cot^{-1}x\)) gives a range of \((-\frac{π}{2},\frac{π}{2}]\) instead of \((0,π)\). For x < 0, always use the identity \(\cot^{-1}x=\frac{π}{2}-\tan^{-1}x\) to compute \(\cot^{-1}x\), so it has a range \((0,π)\) for negative \(x\).

(4) \(\cot^{-1}0=\frac{π}{2},\cot^{-1}1=\frac{π}{4},\cot^{-1}(-1)=\frac{3π}{4}=\frac{\pi}{2}-\tan^{-1}(-1),\cot^{-1}\frac{\sqrt{3}}{3}=\frac{π}{3},\cot^{-1}\frac{-\sqrt{3}}{3}=\frac{2π}{3},\cot^{-1}\sqrt{3}=\frac{π}{6},\cot^{-1}(-\sqrt{3})=\frac{5π}{6}\). Check acot(0) || acot(1) || pi/2-atan(-1) || acot(3**(1/2)/3) || pi/2-atan(-3**(1/2)/3) || acot(3**(1/2)) || pi/2-atan(-3**(1/2)) ||.

(5) \(\sec^{-1}1=0,\sec^{-1}(-1)=π,\sec^{-1}2=\frac{π}{3},\sec^{-1}\sqrt{2}=\frac{π}{4},\sec^{-1}\frac{2\sqrt{3}}{3}=\frac{π}{6},\sec^{-1}(-2)=\frac{2π}{3}\), \(\sec^{-1}(-\sqrt{2})=\frac{3π}{4},\sec^{-1}\frac{-2\sqrt{3}}{3} =\frac{5π}{6}\). Check asec(1) || asec(-1) || asec(2) || asec(-2) || asec(2*3**(1/2)/3) || asec(2**(1/2)) || asec(-2*3**(1/2)/3) || asec(-2**(1/2)) ||.

(6) \(\csc^{-1}1=\frac{π}{2},\csc^{-1}(-1)=\frac{-π}{2},\csc^{-1}2=\frac{π}{6},\csc^{-1}\sqrt{2}=\frac{π}{4},\csc^{-1}\frac{2\sqrt{3}}{3}=\frac{π}{3},\csc^{-1}(-2)=\frac{-π}{6}\),
\(\csc^{-1}(-\sqrt{2})=\frac{-π}{4},\csc^{-1}\frac{-2\sqrt{3}}{3} =\frac{-π}{3}\). Check acsc(1) || acsc(-1) || acsc(2) || acsc(-2) || acsc(2*3**(1/2)/3) || acsc(2**(1/2)) || acsc(-2*3**(1/2)/3) || acsc(-2**(1/2)) ||.

Negative values \(\sin^{-1}(-x)=-\sin^{-1}x,\tan^{-1}(-x)=-\tan^{-1}x,\csc^{-1}(-x)=-\csc^{-1}x\),
\(\cos^{-1}(-x)=π-\cos^{-1}x,\cot^{-1}(-x)=π-\cot^{-1}x,\sec^{-1}(-x)=π-\sec^{-1}x\).

Examples
(1) asin(-x) || atan(-x) || acsc(-x) || acos(-0.2)+acos(0.2) || asec(-5.0)+asec(5.0) ||.
(2) Since \(\cot^{-1}(-2)+\cot^{-1}2=\frac{π}{2}-\tan^{-1}(-2)+\frac{π}{2}-\tan^{-1}2\)\(=π+\tan^{-1}2-\tan^{-1}2=π\). Check pi/2-atan(-2)+pi/2-atan(2) ||. Note that acot(-2)+acot(2) || gives 0 because the module of "acot" function has a range that is different from (\(0,π\)).

Graphs of inverse trigonometric functions Use "plt", "imf" and other graph modules to plot trigonometric functions and their inverses.

Examples
(1) Use "plt" to plot inverse functions such as plt;asin(x) || plt;acos(x) || plt;atan(x) || plt;asec(x) || plt;acsc(x) ||.
(2) Use the identity \(\cot^{-1}x=\frac{π}{2}-\tan^{-1}x\) to graph \(\cot^{-1}x\) by plt;pi/2-atan(x) ||.
(3) Try other transformations of these basic graphs by plt;3*asin(x/2) || plt;3*acos(2*x-1) || plt;4*asin(x/5) || plt;2*acos(-x/4) || plt;3*asin(-x/3) || plt;2*asin(1-x/2)+3 ||.

Reciprocal values \(\sin^{-1}\frac{1}{x}=\csc^{-1}x,\cos^{-1}\frac{1}{x}=\sec^{-1}x,\csc^{-1}\frac{1}{x}=\sin^{-1}x, \sec^{-1}\frac{1}{x} =\cos^{-1}x\) for all \(x\) in their domains. If \(x>0,\tan^{-1}\frac{1}{x}=\cot^{-1}x,\cot^{-1}\frac{1}{x}=\tan^{-1}x,\cot^{-1} \frac{1}{-x}=π-\cot^{-1}\frac{1}{x}\)\(=π+\tan^{-1}(-x)\), and \(\tan^{-1}\frac{1}{-x}=-\cot^{-1}x=\cot^{-1}(-x)-π\).

Examples Check asin(1.0/3)-acsc(3.0) || asin(0.1)==acsc(10.0) || acos(0.2)-asec(5.0) || asin(-0.4)==acsc(-5.0/2) || acos(-1.0/8)==asec(-8.0) || atan(2.0)==acot(0.5) || acot(0.4)==atan(2.5) || atan(0.4)==acot(2.5) || atan(-5.0)+pi/2+atan(-0.2) || pi/2-atan(-5.0)-pi-atan(-0.2) ||.

Examples
(1) \(\sin(\csc^{-1}(x))=\frac{1}{x},\cos(\sec^{-1}(x))=\frac{1}{x},\tan(\cot^{-1}(x))=\frac{1}{x}\). Check sin(acsc(x)) || cos(asec(x)) || tan(acot(x)) || sin(acsc(3)) || cos(asec(-4)) || tan(acot(2)) || tan(acot(-0.5)) ||.
(2) In a similar fashion, \(\csc(\sin^{-1}(x))=\frac{1}{x}\), \(\sec(\cos^{-1}(x))=\frac{1}{x},\cot(\tan^{-1}(x))=\frac{1}{x}\). Check csc(asin(x)) || sec(acos(x)) || cot(atan(x)) || csc(asin(3/5)) || csc(asin(-3/4)) || sec(acos(1/5)) || sec(acos(-3/10)) || cot(atan(-7)) || cot(atan(1/7)) ||.

Complementary angles \(\sin^{-1}x+\cos^{-1}x=\frac{π}{2} \sec^{-1}x+\csc^{-1}x=\frac{π}{2}, \tan^{-1}x+\cot^{-1}x=\frac{π}{2}\).

Examples Check asin(0.1)+acos(0.1) || atan(0.3)+acot(0.3) || asec(3.0)+acsc(3.0) || asin(-0.7)+acos(-0.7) || asec(-5.0)+acsc(-5.0) ||.

Inverse trigonometric identities \(\cos(\sin^{-1}x)=\sin(\cos^{-1}x)=\sqrt{1-x^2},\tan(\sin^{-1}x)=\frac{x}{\sqrt{1-x^2}}\), \( \tan(\cos^{-1}x)=\frac{\sqrt{1-x^2}}{x},\sin(\tan^{-1}x)=\cos(\cot^{-1}x)=\frac{x}{\sqrt{1+x^2}},\cos(\tan^{-1}x)=\sin(\cot^{-1}x)=\frac{1}{\sqrt{1+x^2}}\), \(\cos(\csc^{-1}x)=\sin(\sec^{-1}x)=\sqrt{1-\frac{1}{x^2}},\tan(\csc^{-1}x)=\frac{1}{x\sqrt{1-x^{-2}}},\tan(\sec^{-1}x)=x\sqrt{1-x^{-2}}\)

Examples Check cos(asin(x)) || sin(acos(x)) || tan(asin(x)) || tan(acos(x)) || sin(atan(x)) || cos(pi/2-atan(x)) || sin(atan(x))==cos(pi/2-atan(x)) || cos(atan(x)) || sin(pi/2-atan(x)) || cos(acsc(x)) || sin(asec(x)) || cos(acsc(x))-sin(asec(x)) || tan(acsc(x)) || tan(asec(x)) ||.

Examples cos(asin(1/2)) || cos(asin(-1/3)) || sin(acos(-1/4)) || sin(acos(1/5)) || tan(asin(1/2)) || tan(asin(-1/4)) || tan(acos(-1/3) || tan(acos(2/5)) || sin(atan(2)) || cos(acot(2)) || sin(atan(-1/2)) || cos(pi/2-atan(-1/2)) || cos(atan(5)) || sin(pi/2-atan(5)) || cos(acsc(-8)) || sin(asec(-8)) || cos(acsc(3)) || sin(asec(3)) || tan(acsc(-4)) || tan(acsc(3)) || tan(asec(-4)) || tan(asec(5)) || cot(pi-x) || sin(pi/2+x) || tan(-pi/5) || tan(3*pi/2-x) || sin(acos(x) || 1+cot(x)**2 || cos(pi+x) || csc(2*pi-x) || 3*sin(y)-4*sin(y)**3 || 4*cos(a)**3-3*cos(a) || asin(0.7)+acos(0.7) || tan(asin(s)) || tan(acos(t)) || sin(atan(u)) || cos(acot(v)) || atan(0.5)+atan(1.0/3) ||.

Hyperbolic functions Hyperbolic functions are related to a hyperbola in much the same way as trigonometric (or circular) functions are related to a circle.
\(y=\sinh x=\frac{e^x-e^{-x}}{2},  x∈R, y∈R\)
\(y=\cosh x=\frac{e^x+e^{-x}}{2},   x∈R, y≥1\)
\(y=\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}},  x∈R, |y|< 1\)
\(y=\coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}},   x≠0, |y| > 1\)
\(y=\) sech \(x=\frac{1}{\cosh x}=\frac{1}{e^x+e^{-x}},   x∈R, 0< y≤ 1\)
\(y=\) csch \(x=\frac{1}{\sinh x}=\frac{1}{e^x-e^{-x}},   x≠0, y∈R\).

Increasing and decreasing The functions \(\sinh x\) and \(\tanh x\) are increasing for all \(x\); their reciprocals csch \(x\) and \(\coth x\) are decreasing for all \(x≠0\). These four functions are monotonic and one-to-one.

The functions \(\cosh x\) and sech \(x\) are not one-to-one, they are all positive and symmetric about the y-axis for all \(x\). For \(x>0,\cosh x\) is increasing and sech \(x\) is decreasing; for \(x< 0, \cosh x\) is decreasing and sech \(x\) is increasing. The point (0, 1) is the minimum point on the graph of \(\cosh x\) and the maximum point on the graph of sech \(x\).

Examples sinh(2)>sinh(1) || sinh(-3)>sinh(-4) || sinh(0.5)>sinh(0) || cosh(3)>cosh(2) || cosh(-3)>cosh(-2) || tanh(8)>tanh(5) || tanh(0)>tanh(-1) || coth(5)>coth(7) || coth(-9)>coth(-6) || csch(8)>csch(9) || csch(-7)>csch(-5) || sech(9)>sech(10) || sech(0)>sech(-3) ||.

Even and odd Just as trigonometric functions, the functions \(\cosh x\) and \(\text{sech }x\) are even functions, and the rest four are odd functions. Thus, \(\sinh(-x)=-\sinh x,\tanh(-x)=-\tanh x,\coth(-x)=-\coth x,\text{csch}(-x)\)\(=-\text{csch }x\), and \(\cosh(-x)=\cosh x,\text{sech}(-x)=\text{sech } x\). Check sinh(-x) || cosh(-x) || tanh(-x) || coth(-x) || sech(-x) || csch(-x) ||.

Graphs of hyperbolic functions You use the "plt" and other modules to graph hyperbolic functions.

Examples Check plt;sinh(x) || plt;cosh(x) || plt;tanh(x) || plt;coth(x) || plt;sech(x) || plt;csch(x) ||. Check the transformations of these basic graphs by plt;2*sinh(x/3) || plt;-cosh(x-3) || plt;3*tanh(-x) || plt;-sinh(2*x-3)/4 ||.

Evaluating hyperbolic functions Most values of hyperbolic functions are irrational. Exact values are possible if \(x\) is in form of natural logarithm. Replacing \(x\) by \(\ln x\) in the defining equations, we have exact values \(\sinh\ln x=\frac{e^{\ln x}-e^{-\ln x}}{2}=\frac{x-x^{-1}}{2}=\frac{x^2-1}{2x}, \cosh\ln x=\frac{x+x^{-1}}{2},  \tanh\ln x=\frac{x-x^{-1}}{x+x^{-1}}\),
\(\coth\ln x=\frac{x+x^{-1}}{x-x^{-1}},\) sech \(\ln x=\frac{2}{x+x^{-1}},  \text{csch}\ln x=\frac{2}{x-x^{-1}}\).

As a result, \(\sinh(\ln2)=\frac{2-0.5}{2}=0.75,\cosh(\ln2)=\frac{2+0.5}{2}=1.25\).

Examples Check sinh(log(2.0)) || cosh(log(2.0)) || tanh(log(2.0)) || coth(log(2.0)) || sech(log(2.0)) || csch(log(2.0)) || sinh(-log(5.0)) || cosh(-log(5.0)) || tanh(-log(5.0)) || coth(-log(5.0)) || sinh(0) || tanh(0) || cosh(0) || sech(0) || sech(-log(5.0)) || csch(-log(5.0)) || sinh(log(0.4)) || cosh(log(0.4)) || tanh(log(0.4)) || coth(log(0.4)) || sech(log(0.6)) || csch(log(0.4)) ||.

Peroperties of hyperbolic functions   \(\sinh x+\cosh x=e^x,\cosh(x)-\sinh x=e^{-x},\cosh^2x-\sinh^2x=1\), \(\tanh^2x+\text{sech}^2x=1,\coth^2x-\text{csch}^2x=1\).

Examples sinh(x)+cosh(x) || cosh(x)-sinh(x) || sinh(1)+cosh(1) || sinh(-1)+cosh(-1) || sinh(-2)+cosh(-2) || sinh(4)+cosh(4) || cosh(3)-sinh(3) ||cosh(-5)-sinh(-5) || cosh(-1)-sinh(-1) || cosh(x)**2-sinh(x)**2 || cosh(2)**2-sinh(2)**2 || cosh(-3)**2-sinh(-3)**2 || cosh(0.6)**2-sinh(0.6)**2 || tanh(x)**2+sech(x)**2 || tanh(6)**2+sech(6)**2 || tanh(-5)**2+sech(-5)**2 || coth(x)**2-csch(x)**2 || coth(4)**2-csch(4)**2 || coth(-7)**2-csch(-7)**2 ||.

Sum of arguments \(\sinh(x\pm y)=\sinh x\cosh y\pm \cosh x\sinh y,\cosh(x\pm y)=\cosh x\cosh y\pm \sinh x\sinh y\),
\(\tanh(x\pm y)=\frac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}\).

Examples sinh(x+y)==sinh(x)*cosh(y)+cosh(x)*sinh(y) || sinh(x-y)==sinh(x)*cosh(y)-cosh(x)*sinh(y) || cosh(x+y)==cosh(x)*cosh(y)+sinh(x)*sinh(y) || cosh(x-y)==cosh(x)*cosh(y)-sinh(x)*sinh(y) || tanh(x+y)==(tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)) || tanh(x-y)==(tanh(x)-tanh(y))/(1-tanh(x)*tanh(y)) || sinh(2)*cosh(3)+cosh(2)*sinh(3) || cosh(2)*cosh(1)-sinh(2)*sinh(1) || (tanh(4)-tanh(3.0))/(1-tanh(4)*tanh(3))-tanh(1) ||.

Double arguments \(\sinh 2x=2\sinh x\cosh x, \cosh 2x=\cosh^2x+\sinh^2x, \tanh2x=\frac{2\tanh x}{1+\tanh^2x}\). These formulas can be obtained by replacing \(y\) by \(x\) in the previous formulas for sum of arguments.

Examples Check sinh(2*x)==2*sinh(x)*cosh(x) || cosh(2*x)==cosh(x)**2+sinh(x)**2 || tanh(2*x)==2*tanh(x)/(1+tanh(x)**2) || 2*sinh(2)*cosh(2) || cosh(3)**2+sinh(3)**2 || 2*tanh(5)/(1+tanh(5)**2) ||.

Half arguments \(\sinh^2x=\frac{\cosh 2x-1}{2}, \cosh^2x=\frac{\cosh 2x+1}{2}, \tanh\frac{x}{2}=\frac{\cosh x-1}{\sinh x}=\frac{\sinh x}{1+\cosh x}\).

Examples Check sinh(x)**2==(cosh(2*x)-1)/2 || cosh(x)**2-(cosh(2*x)+1)/2 || tanh(1)-(cosh(2)-1)/sinh(2) || tanh(1)-cosh(2)/(1+sinh(2)) ||.

Sum to product \(\sinh x+\sinh y=2\sinh\frac{x+y}{2}\cosh\frac{x-y}{2},\sinh x-\sinh y=2\sinh\frac{x-y}{2}\cosh\frac{x+y}{2}\), \(\cosh x+\cosh y=2\cosh\frac{x+y}{2}\cosh\frac{x-y}{2},\cosh x-\cosh y=2\sinh\frac{x+y}{2}\sinh\frac{x-y}{2}\).

Examples Check 2*sinh((x+y)/2)*cosh((x-y)/2) || 2*sinh((x+y)/2)*sinh((x-y)/2) || 2*cosh((x+y)/2)*cosh((x-y)/2) || 2*cosh((x+y)/2)*sinh((x-y)/2) ||.

Other formulas \((\sinh x + \cosh x)^r=(e^{x})^r=e^{rx}=\sinh rx+\cosh rx,  (\cosh x-\sinh x)^r=e^{-rx}\)\(=\sinh rx-\cosh rx\), \(\cosh3x=4\cosh^3x-3\cosh x\).

Examples Check (sinh(x)+cosh(x))**2 || (sinh(x)+cosh(x))**3 || (sinh(x)+cosh(x))**(-1) || (sinh(x)+cosh(x))**(-3) || (cosh(x)-sinh(x))**(-4) || (cosh(x)-sinh(x))**5 || (cosh(x)-sinh(x))**(-10) || cosh(3*x)==4*cosh(x)**3-3*cosh(x) ||.

Inverse hyperbolic functions Hyperbolic functions are not periodic, but \(\cosh x\) and sech \(x\) are even and not one-to-one. So their inverses are defined in some restricted domains. Note the domains and ranges of these inverse hyperbolic functions.

\(y=\sinh^{-1}x=\ln(x+\sqrt{x^2+1})⇔ \sinh y=x, x∈R, y∈R\)
\(y=\cosh^{-1}x=\ln(x+\sqrt{x^2-1})⇔ \sinh y=x,  x≥1, y≥ 0\)
\(y=\tanh^{-1}x=\frac{1}{2}\ln\frac{1+x}{1-x}⇔ \tanh y=x,  |x|< 1, y∈R\)
\(y=\coth^{-1}x=\tanh^{-1}\frac{1}{x}=\frac{1}{2}\ln\frac{1+x}{x-1}⇔ \coth y=x,  |x|> 1, y≠0\)
\(y=\text{sech}^{-1}x=\cosh^{-1}\frac{1}{x}=\ln(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1})⇔ \text{sech }y=x,  0< x≤1, y≥0\)
\(y=\text{csch}^{-1}x=\sinh^{-1}\frac{1}{x}=\ln(\frac{1}{x}+\sqrt{\frac{1}{x^2}+1})⇔\text{csch }y=x,  x≠0, y≠0\)

Since \(\sinh\ln x=\frac{x^2-1}{2x}, \cosh\ln x=\frac{x^2+1}{2x},\tanh\ln x=\frac{x^2-1}{x^2+1}\), we have \(\ln x=\sinh^{-1}(\frac{x^2-1}{2x}) =\cosh^{-1}(\frac{x^2+1}{2x})\) \(=\tanh^{-1}(\frac{x^2-1}{x^2+1})\).

Examples (1) Check log(2.0)==asinh(0.75) || log(2.0)==acosh(5.0/4) || log(2.0)==atanh(0.6) ||. (2) By reciprocal relations, \(\ln x=\text{csch}^{-1}(\frac{2x}{x^2-1})=\text{sech}^{-1}(\frac{2x}{x^2+1})=\tanh^{-1}(\frac{x^2+1}{x^2-1})\). Check log(2.0)==acsch(4.0/3) || log(2.0)==asech(4/5.0) || log(2.0)-acoth(5.0/3) ||.

Examples
(1) \(\sinh^{-1}(2)=\ln(2+\sqrt{5}),\sinh^{-1}(-1)=\ln(\sqrt{2}-1)\) by log(2+5**(1/2))==asinh(2) || log(-1+2**(1/2))-asinh(-1).
(2) \(\cosh^{-1}(2)=\ln(2+\sqrt{3})\) by acosh(2)==log(2+3**(1/2)) ||.
(3) \(\tanh^{-1}(\frac{1}{2})=\frac{1}{2}\ln3\) by atanh(0.5)==log(3.0)/2 ||.
(4) \(\coth^{-1}(3)=\frac{1}{2}\ln2\) by acoth(3.0)-log(2.0)/2 ||.
(5) csch\(^{-1}(\frac{1}{4})=\ln(4+\sqrt{17})\) by acsch(0.25)-log(4.0+17**0.5) ||.
(6) sech\(^{-1}(\frac{1}{5})=\ln(5+\sqrt{24})\) by asech(0.2)-log(5.0+24**0.5) ||.

Negative arguments Since \(\cosh^{-1}x\) and sech\(^{-1}x\) have positive domains and ranges, negative arguments are not defined. The rest four are odd functions, or \(\sinh^{-1}(-x)=-\sinh^{-1}x, \tanh^{-1}(-x)=-\tanh^{-1}x,\coth^{-1}(-x)\)\(=-\coth^{-1}x\) and csch\(^{-1}(-x)=-\) csch\(^{-1}x\).

Examples Check asinh(-x) || atanh(-x) || acoth(-x) || acsch(-x) || asinh(-2)==-asinh(2) || atanh(-0.5)==-atanh(0.5) || acoth(-3)==-acoth(3) || acsch(-4)==-acsch(4) ||.

Graphs of inverse hyperbolic functions Use "plt" module we can obtain the basic graphs of inverse hyperbolic functions and their transformations.

Examples plt;asinh(x);sinh(x) || plt;acosh(x);cosh(x) || plt;atanh(x);tanh(x) || plt;acoth(x);coth(x) || plt;asech(x);sech(x) || plt;acsch(x);csch(x) || plt;2*asinh(3*x-1) || plt;-2*acosh(x/3+2) || plt;3*atanh(1.5*x-2)-3 ||.

Composition \(\sinh(\cosh^{-1}x)=\sqrt{x^2-1}, \cosh(\sinh^{-1}x)=\sqrt{x^2+1}, \sinh(\tanh^{-1}x)=\frac{x}{\sqrt{1-x^2}},\)
\(\tanh(\sinh^{-1}x)=\frac{x}{\sqrt{x^2+1}}, \cosh(\tanh^{-1}x)=\frac{1}{\sqrt{1-x^2}}, \tanh(\cosh^{-1}x)=\frac{\sqrt{x^2-1}}{x}\).

Examples
(1) plt;sinh(acosh(x)) || plt;cosh(asinh(x)) || plt;sinh(atanh(x)) || plt;tanh(asinh(x)) || plt;cosh(atanh(x)) || plt;tanh(acosh(x)) ||.
(2) \(\sinh(\cosh^{-1}(\sqrt{5}))=2\) by sinh(acosh(5**(1/2))) ||.
(3) \(\cosh(\sinh^{-1}(\sqrt{8}))=3\) by cosh(asinh(8**(1/2))) ||.
(4) \(\tanh(\sinh^{-1}(1))=\frac{\sqrt{2}}{2}\) by tanh(asinh(1.0))-2**0.5/2 ||.