Trigonometric Equations

Plot of the six trigonometric functions, the unit circle, and a line for the angle $\theta =0.7$ radians. The points labelled Template:Color, Template:Color, Template:Color represent the length of the line segment from the origin to that point. Template:Color, Template:Color, and Template:Color are the heights to the line starting from the $x$-axis, while Template:Color, Template:Color, and Template:Color are lengths along the $x$-axis starting from the origin.

The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are $\sin(\theta ),\cos(\theta ),$ and $\tan(\theta ),$ respectively, where $\theta$ denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., $\sin \theta$ and $\cos \theta ,$ if an interpretation is unambiguously possible.

The sine of an angle is defined, in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse).

$\sin \theta ={\frac {\text{opposite}}{\text{hypotenuse}}}.$

The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse.

$\cos \theta ={\frac {\text{adjacent}}{\text{hypotenuse}}}.$

The tangent of an angle in this context is the ratio of the length of the side that is opposite to the angle divided by the length of the side that is adjacent to the angle. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of $\sin$ and $\cos$ from above:

$\tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {\text{opposite}}{\text{adjacent}}}.$

The remaining trigonometric functions secant ($\sec$), cosecant ($\csc$), and cotangent ($\cot$) are defined as the reciprocal functions of cosine, sine, and tangent, respectively. Rarely, these are called the secondary trigonometric functions:

$\sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }},\quad \cot \theta ={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}.$

These definitions are sometimes referred to as ratio identities.

Inverse trigonometric functions

The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine ($\sin ^{-1}$) or arcsine (arcsin or asin), satisfies

$$\sin(\arcsin x)=x\quad {\text{for}}\quad |x|\leq 1$$

and

$$\arcsin(\sin x)=x\quad {\text{for}}\quad |x|\leq {\frac {\pi }{2}}.$$

This article will denote the inverse of a trigonometric function by prefixing its name with "$\operatorname {arc}$". The notation is shown in the table below.

Compositions of trig and inverse trig functions

$${\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}$$

Taking the multiplicative inverse of both sides of the each equation above results in the equations for $\csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ and }}\cot ={\frac {1}{\tan }}.$ The right hand side of the formula above will always be flipped. For example, the equation for $\cot(\arcsin x)$ is:

$$\cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}$$

while the equations for $\csc(\arccos x)$ and $\sec(\arccos x)$ are:

$$\csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ and }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.$$

Solutions to elementary trigonometric equations

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values $\theta ,r,s,x,$ and $y$ all lie within appropriate ranges so that the relevant expressions below are well-defined. In the table below, "for some $k\in \mathbb {Z}$" is just another way of saying "for some integer $k.$"

Equation	if and only if	Solution						where...
$\sin \theta =y$	$\iff$	$\theta =\,$	$(-1)^{k}$	$\arcsin(y)$	$+$		$\pi k$	for some $k\in \mathbb {Z}$
$\csc \theta =r$	$\iff$	$\theta =\,$	$(-1)^{k}$	$\operatorname {arccsc}(r)$	$+$		$\pi k$	for some $k\in \mathbb {Z}$
$\cos \theta =x$	$\iff$	$\theta =\,$	$\pm \,$	$\arccos(x)$	$+$	$2$	$\pi k$	for some $k\in \mathbb {Z}$
$\sec \theta =r$	$\iff$	$\theta =\,$	$\pm \,$	$\operatorname {arcsec}(r)$	$+$	$2$	$\pi k$	for some $k\in \mathbb {Z}$
$\tan \theta =s$	$\iff$	$\theta =\,$		$\arctan(s)$	$+$		$\pi k$	for some $k\in \mathbb {Z}$
$\cot \theta =r$	$\iff$	$\theta =\,$		$\operatorname {arccot}(r)$	$+$		$\pi k$	for some $k\in \mathbb {Z}$

For example, if $\cos \theta =-1$ then $\theta =\pi +2\pi k=-\pi +2\pi (1+k)$ for some $k\in \mathbb {Z} .$ While if $\sin \theta =\pm 1$ then $\theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)$ for some $k\in \mathbb {Z} ,$ where $k$ is even if $\sin \theta =1$; odd if $\sin \theta =-1.$ The equations $\sec \theta =-1$ and $\csc \theta =\pm 1$ have the same solutions as $\cos \theta =-1$ and $\sin \theta =\pm 1,$ respectively. In all equations above Template:Em for those just solved (i.e. except for $\sin$/$\csc \theta =\pm 1$ and $\cos$/$\sec \theta =-1$), for fixed $r,s,x,$ and $y,$ the integer $k$ in the solution's formula is uniquely determined by $\theta .$

Other identities

These inverse trigonometric functions are related to one another by the formulas:

$${\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}$$

which hold whenever they are well-defined (that is, whenever $x,r,s,-x,-r,{\text{ and }}-s$ are in the domains of the relevant functions).

Aslo,

$${\begin{aligned}\arcsin x+\arccos x&={\dfrac {\pi }{2}}\\\arctan x+\operatorname {arccot} x&={\dfrac {\pi }{2}}\\\end{aligned}}$$

$${\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{if }}x>0\\-{\frac {\pi }{2}},&{\text{if }}x<0\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{if }}x>0\\{\frac {3\pi }{2}},&{\text{if }}x<0\end{cases}}\\\end{aligned}}$$

$$\arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ and }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x$$

$$\arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ and }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x$$

$$\arctan {\frac {1}{x}}=\arctan {\frac {1}{x+y}}+\arctan {\frac {y}{x^{2}+xy+1}}$$

Resources

• Trigonometric Equations. Written notes created by Professor Esparza, UTSA.

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Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.