Section 4.2 Trigonometry
Subsection 4.2.1 Trigonometric Functions
If \((x,y)\) are the coordinates of a point on the unit circle and if \(\theta\) is the angle for that point (measured counterclockwise from the positive \(x\) axis), then the basic trigonometric function take the angle \(\theta\) as their input and output the coordinates \(x\) and \(y\) (in some combination).
\begin{align*}
\sin \theta \amp = y \\
\cos \theta \amp = x \\
\tan \theta \amp = \frac{\sin \theta}{\cos \theta} =
\frac{y}{x}\\
\sec \theta \amp = \frac{1}{\cos \theta} = \frac{1}{x} \\
\csc \theta \amp = \frac{1}{\sin \theta} = \frac{1}{y}\\
\cot \theta \amp = \frac{\cos \theta}{\sin \theta} =
\frac{x}{y}
\end{align*}
Subsection 4.2.2 Unit Circle
A unit circle diagram, presented here, gives the trig values of several commonly used angles. All angles here are measured in radians. Remember, when using this diagram, that sine gives the \(y\) coordinates and cosine gives the \(x\) coordinate.
Subsection 4.2.3 Squares
\begin{align*}
\sin^2 x + \cos^2 x \amp = 1 \\
1 + \tan^2 x \amp = \sec^2 x \\
1 + \cot^2 x \amp = \csc^2 x
\end{align*}
Subsection 4.2.4 Symmetry
\begin{align*}
\sin (-x) \amp = -\sin x \\
\cos (-x) \amp = \cos x \\
\tan (-x) \amp = - \tan x
\end{align*}
Subsection 4.2.5 Shifts
\begin{align*}
\sin \left(x + \dfrac{\pi}{2} \right) \amp = \cos x \\[0.8em] \\
\cos \left(x - \dfrac{\pi}{2} \right) \amp = \sin x \\[0.8em] \\
\tan \left( \dfrac{\pi}{2} - x \right) \amp = \cot x \\[0.8em]
\end{align*}
Subsection 4.2.6 Additions
\begin{align*}
\sin (x + y) \amp = \sin x \cos y + \cos x \sin y \\
\cos (x + y) \amp = \cos x \cos y - \sin x \sin y \\
\tan (x + y) \amp = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}
\end{align*}
Subsection 4.2.7 Subtractions
\begin{align*}
\sin (x - y) \amp = \sin x \cos y - \cos x \sin y \\
\cos (x - y) \amp = \cos x \cos y + \sin x \sin y \\
\tan (x - y) \amp = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}
\end{align*}
Subsection 4.2.8 Double Angles
\begin{align*}
\sin 2x \amp = 2 \sin x \cos x \\
\cos 2x \amp = \cos^2 x - \sin^2 x
\end{align*}
Subsection 4.2.9 Half Angles
\begin{align*}
\sin^2 x \amp = \dfrac{1 - \cos 2x }{2} \\
\cos^2 x \amp = \dfrac{1 + \cos 2x }{2}
\end{align*}
Subsection 4.2.10 Sum to Product
\begin{align*}
\sin x + \sin y \amp = 2 \sin \left( \dfrac{x+y}{2} \right)
\cos \left( \dfrac{x-y}{2} \right)\\
\cos x + \cos y \amp = 2 \cos \left( \dfrac{x+y}{2} \right)
\cos \left( \dfrac{x-y}{2} \right) \\
\sin x - \sin y \amp = 2 \sin \left( \dfrac{x-y}{2} \right)
\cos \left( \dfrac{x+y}{2} \right)\\
\cos x - \cos y \amp = -2 \sin \left( \dfrac{x+y}{2} \right)
\sin \left( \dfrac{x-y}{2} \right)
\end{align*}
Subsection 4.2.11 Product to Sum
\begin{align*}
\sin x \sin y \amp = \dfrac{\cos (x-y) - \cos (x+y)}{2} \\
\cos x \cos y \amp = \dfrac{\cos (x-y) + \cos (x+y)}{2} \\
\sin x \cos y \amp = \dfrac{\sin (x+y) + \sin (x-y)}{2} \\
\cos x \sin y \amp = \dfrac{\sin (x+y) - \sin (x-y)}{2}
\end{align*}
Subsection 4.2.12 Sine Law
\begin{align*}
\dfrac{\sin A}{a} \amp = \dfrac{\sin B}{b} \ \ = \ \
\dfrac{\sin C}{c}
\end{align*}
Subsection 4.2.13 Cosine Law
\begin{align*}
c^2 \amp = a^2 + b^2 - ab\cos c
\end{align*}
Subsection 4.2.14 Superposition
\begin{align*}
a \sin x + b \sin x \amp = \sqrt{a^2 + b^2} \sin (x + \phi)
\amp \text{ where } \amp \phi = \arcsin \left(
\dfrac{b}{\sqrt{a^2+b^2}} \right)
\end{align*}
Subsection 4.2.15 Inverse Trigonometric Functions
To invert trig functions, we have to restrict the domains, since the functions are not monotonic on their domains. There are many choices for these restrictions, but there are some standard choices which are presented in the follow table. The inverse trig functions are written with an arc prefix. (This is strongly preferred to writing \(\sin^{-1}(x)\) and similar notations, since the superscript \((-1)\) in these notations is not an exponent and does not agree with normal exponential notation for trig functions, such as \(\sin^2
(x)\)). With these restrictions, the range of the original function becomes the domain of the inverse function.
\begin{align*}
\amp \text{Function} \amp \amp \text{Domain} \amp \amp
\text{Range} \amp \amp \text{Inverse} \\
\amp \sin x \amp \amp \left[ \frac{-\pi}{2}, \frac{\pi}{2}
\right] \amp \amp [-1,1] \amp \amp \arcsin x\\
\amp \cos x \amp \amp [0, \pi] \amp \amp [-1,1] \amp \amp
\arccos x \\
\amp \tan x \amp \amp \left( \frac{-\pi}{2}, \frac{\pi}{2}
\right) \amp \amp \RR \amp \amp \arctan x\\
\amp \sec x \amp \amp \left[ 0, \frac{\pi}{2} \right) \amp
\amp [1, \infty) \amp \amp \arcsec x\\
\amp \csc x \amp \amp \left( 0, \frac{\pi}{2} \right] \amp
\amp [1, \infty) \amp \amp \arccsc x\\
\amp \cot x \amp \amp (0, \pi) \amp \amp \RR \amp \amp
\arccot x
\end{align*}
