A.1: Trigonometry

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A.1.1 Trigonometry — Graphs

A.1.2 Trigonometry — Special Triangles

$special_triangles.svg$

From the above pair of special triangles we have

\[\begin{align*} \sin \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \sin \frac{\pi}{6} &= \frac{1}{2} & \sin \frac{\pi}{3} &= \frac{\sqrt{3}}{2}\\ \cos \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \cos \frac{\pi}{6} &= \frac{\sqrt{3}}{2} & \cos \frac{\pi}{3} &= \frac{1}{2}\\ \tan \frac{\pi}{4} &= 1 & \tan \frac{\pi}{6} &= \frac{1}{\sqrt{3}} & \tan \frac{\pi}{3} &= \sqrt{3} \end{align*}\]

A.1.3 Trigonometry — Simple Identities

Periodicity
\[\begin{align*} \sin(\theta+2\pi) &= \sin(\theta) & \cos(\theta+2\pi) &= \cos(\theta) \end{align*}\]
Reflection
\[\begin{align*} \sin(-\theta)&=-\sin(\theta) & \cos(-\theta) &=\cos(\theta) \end{align*}\]
Reflection around $\pi/4$
\[\begin{align*} \sin\left(\tfrac{\pi}{2}-\theta\right)&=\cos\theta & \cos\left(\tfrac{\pi}{2}-\theta\right)&=\sin\theta \end{align*}\]
Reflection around $\pi/2$
\[\begin{align*} \sin\left(\pi-\theta\right)&=\sin\theta & \cos\left(\pi-\theta\right)&=-\cos\theta \end{align*}\]
Rotation by $\pi$
\[\begin{align*} \sin\left(\theta+\pi\right)&=-\sin\theta & \cos\left(\theta+\pi\right)&=-\cos\theta \end{align*}\]
Pythagoras
\[\begin{align*} \sin^2\theta + \cos^2 \theta &=1\\ \tan^2\theta + 1 &= \sec^2\theta\\ 1 + \cot^2 \theta &=\csc^2\theta \end{align*}\]
$\sin$ and $\cos$ building blocks
\[\begin{gather*} \tan\theta=\frac{\sin\theta}{\cos\theta}\quad \csc\theta=\frac{1}{\sin\theta}\quad \sec\theta=\frac{1}{\cos\theta}\quad \cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{1}{\tan\theta} \end{gather*}\]

A.1.4 Trigonometry — Add and Subtract Angles

Sine
\[\begin{align*} \sin(\alpha \pm \beta) &= \sin(\alpha )\cos(\beta) \pm \cos(\alpha )\sin(\beta) \end{align*}\]
Cosine
\[\begin{align*} \cos(\alpha \pm \beta) &= \cos(\alpha )\cos(\beta) \mp \sin(\alpha )\sin(\beta) \end{align*}\]
Tangent
\[\begin{align*} \tan(\alpha +\beta)&=\frac{\tan\alpha +\tan\beta}{1-\tan\alpha \tan\beta}\\ \tan(\alpha -\beta)&=\frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta} \end{align*}\]
Double angle
\[\begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\ \cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta)\\ &= 2\cos^2(\theta) - 1\\ &= 1 - 2\sin^2(\theta)\\ \tan(2\theta) &= \frac{2\tan(\theta)}{1-\tan^2\theta}\\ \cos^2\theta&=\frac{1+\cos(2\theta)}{2}\\ \sin^2\theta&=\frac{1-\cos(2\theta)}{2}\\ \tan^2\theta&=\frac{1-\cos(2\theta)}{1+\cos(2\theta)} \end{align*}\]
Products to sums
\[\begin{align*} \sin(\alpha )\cos(\beta)&= \frac{\sin(\alpha +\beta) + \sin(\alpha -\beta)}{2}\\ \sin(\alpha )\sin(\beta)&= \frac{\cos(\alpha -\beta) - \cos(\alpha +\beta)}{2}\\ \cos(\alpha )\cos(\beta)&= \frac{\cos(\alpha -\beta) + \cos(\alpha +\beta)}{2} \end{align*}\]
Sums to products
\[\begin{align*} \sin\alpha +\sin\beta &= 2 \sin\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2}\\ \sin\alpha -\sin\beta &= 2 \cos\frac{\alpha +\beta}{2}\sin\frac{\alpha -\beta}{2}\\ \cos\alpha +\cos\beta &= 2 \cos\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2}\\ \cos\alpha -\cos\beta &= -2 \sin\frac{\alpha +\beta}{2}\sin\frac{\alpha -\beta}{2} \end{align*}\]

A.1.5 Inverse Trigonometric Functions

\[ \arcsin x \nonumber \]

Domain: $-1 \leq x \leq 1$

Range: $-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}$

\[ \arccos x \nonumber \]

Domain: $-1 \leq x \leq 1$

Range: $0 \leq \arccos x \leq \pi$

$\arctan x \nonumber$

Domain: all real numbers

Range: $-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}$

Since these functions are inverses of each other we have

\[\begin{align*} \arcsin(\sin \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\\ \arccos(\cos \theta) &= \theta & 0 \leq \theta \leq \pi\\ \arctan(\tan \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \end{align*}\]

and also

\[\begin{align*} \sin(\arcsin x) &= x & -1 \leq x \leq 1\\ \cos(\arccos x) &= x & -1 \leq x \leq 1\\ \tan(\arctan x) &= x & \text{any real } x \end{align*}\]

\[ \textrm{arccsc} x \nonumber \]

Domain: $|x|\ge 1$

Range: $-\frac{\pi}{2} \leq \textrm{arccsc} x \leq \frac{\pi}{2}$

\[ \textrm{arccsc} x \ne 0 \nonumber \]

\[ \textrm{arcsec} x \nonumber \]

Domain: $|x|\ge 1$

Range: $0 \leq \textrm{arcsec} x \leq \pi$

\[ \textrm{arcsec} x \ne \frac{\pi}{2} \nonumber \]

\[ \textrm{arccot} x \nonumber \]

Domain: all real numbers

Range: $0 \lt \textrm{arccot} x \lt \pi$

Again

\[\begin{align*} \textrm{arccsc}(\csc \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2},\ \theta\ne 0\\ \textrm{arcsec}(\sec \theta) &= \theta & 0 \leq \theta \leq \pi,\ \theta\ne \frac{\pi}{2}\\ \textrm{arccot}(\cot \theta) &= \theta & 0 \lt \theta \lt \pi \end{align*}\]

and

\[\begin{align*} \csc(\textrm{arccsc} x) &= x & |x|\ge 1\\ \sec(\textrm{arcsec} x) &= x & |x|\ge 1\\ \cot(\textrm{arccot} x) &= x & \text{any real } x \end{align*}\]

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