Fundamental Trigonometric Identities
An identity is an equation that is true for all values of the variable(s). A trigonometric identity is an identity that involves trigonometric functions. The fundamental trigonometric identities are as follows.
Reciprocal Identities
\[
\begin{aligned}
\csc x= & \frac{1}{\sin x} \quad \sec x=\frac{1}{\cos x} \quad \cot x=\frac{1}{\tan x} \\
& \tan x=\frac{\sin x}{\cos x} \quad \cot x=\frac{\cos x}{\sin x}
\end{aligned}
\]
Pythagorean Identities
\[
\begin{aligned}
\sin ^{2} x+\cos ^{2} x & =1 \\
\tan ^{2} x+1 & =\sec ^{2} x \\
1+\cot ^{2} x & =\csc ^{2} x
\end{aligned}
\]
Even-Odd Identities
\[
\begin{aligned}
& \sin (-x)=-\sin x \\
& \cos (-x)=\cos x \\
& \tan (-x)=-\tan x
\end{aligned}
\]
Cofunction Identities
\[
\begin{gathered}
\sin \left(\frac{\pi}{2}-x\right)=\cos x \quad \tan \left(\frac{\pi}{2}-x\right)=\cot x \\
\sec \left(\frac{\pi}{2}-x\right)=\csc x \\
\cos \left(\frac{\pi}{2}-x\right)=\sin x \quad \cot \left(\frac{\pi}{2}-x\right)=\tan x \\
\csc \left(\frac{\pi}{2}-x\right)=\sec x
\end{gathered}
\]
Proving Trigonometric Identities
To prove that a trigonometric equation is an identity, we use the following guidelines.
- Start with one side. Select one side of the equation.
- Use known identities. Use algebra and known identities to change the side you started with into the other side.
- Convert to sines and cosines. Sometimes it is helpful to convert all functions in the equation to sines and cosines.
These identities involve the trigonometric functions of a sum or a difference.
\[
\begin{aligned}
& \sin (s+t)=\sin s \cos t+\cos s \sin t \\
& \sin (s-t)=\sin s \cos t-\cos s \sin t
\end{aligned}
\]
\[
\begin{aligned}
& \cos (s+t)=\cos s \cos t-\sin s \sin t \\
& \cos (s-t)=\cos s \cos t+\sin s \sin t
\end{aligned}
\]
\[
\begin{aligned}
& \tan (s+t)=\frac{\tan s+\tan t}{1-\tan s \tan t} \\
& \tan (s-t)=\frac{\tan s-\tan t}{1+\tan s \tan t}
\end{aligned}
\]
Sums of Sines and Cosines
If \(A\) and \(B\) are real numbers, then
\[
A \sin x+B \cos x=k \sin (x+\phi)
\]
where \(k=\sqrt{A^{2}+B^{2}}\) and \(\phi\) satisfies
\[
\cos \phi=\frac{A}{\sqrt{A^{2}+B^{2}}} \quad \sin \phi=\frac{B}{\sqrt{A^{2}+B^{2}}}
\]
These identities involve the trigonometric functions of twice the variable.
\[
\sin 2 x=2 \sin x \cos x
\]
\[
\begin{aligned}
\cos 2 x & =\cos ^{2} x-\sin ^{2} x \\
& =1-2 \sin ^{2} x \\
& =2 \cos ^{2} x-1
\end{aligned}
\]
\[
\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}
\]
These formulas allow us to write a trigonometric expression involving even powers of sine and cosine in terms of the first power of cosine only.
\[
\begin{gathered}
\sin ^{2} x=\frac{1-\cos 2 x}{2} \quad \cos ^{2} x=\frac{1+\cos 2 x}{2} \\
\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}
\end{gathered}
\]
These formulas involve trigonometric functions of half an angle.
\[
\begin{gathered}
\sin \frac{u}{2}= \pm \sqrt{\frac{1-\cos u}{2}} \quad \cos \frac{u}{2}= \pm \sqrt{\frac{1+\cos u}{2}} \\
\tan \frac{u}{2}=\frac{1-\cos u}{\sin u}=\frac{\sin u}{1+\cos u}
\end{gathered}
\]
These formulas involve products and sums of trigonometric functions.
\[
\begin{aligned}
\sin u \cos v & =\frac{1}{2}[\sin (u+v)+\sin (u-v)] \\
\cos u \sin v & =\frac{1}{2}[\sin (u+v)-\sin (u-v)] \\
\cos u \cos v & =\frac{1}{2}[\cos (u+v)+\cos (u-v)] \\
\sin u \sin v & =\frac{1}{2}[\cos (u-v)-\cos (u+v)]
\end{aligned}
\]
\[
\begin{aligned}
& \sin x+\sin y=2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \\
& \sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \\
& \cos x+\cos y=2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} \\
& \cos x-\cos y=-2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}
\end{aligned}
\]
Trigonometric Equations
A trigonometric equation is an equation that contains trigonometric functions. A basic trigonometric equation is an equation of the form \(T(\theta)=c\), where \(T\) is a trigonometric function and \(c\) is a constant. For example, \(\sin \theta=0.5\) and \(\tan \theta=2\) are basic trigonometric equations. Solving any trigonometric equation involves solving a basic trigonometric equation.
If a trigonometric equation has a solution, then it has infinitely many solutions.
To find all solutions, we first find the solutions in one period and then add integer multiples of the period.
We can sometimes use trigonometric identities to simplify a trigonometric equation.