Trigonometry

Section A.1 Trigonometry

Subsection A.1.1 Trigonometry — Graphs

\begin{equation*} \sin \theta \end{equation*}

\begin{equation*} \cos \theta \end{equation*}

\begin{equation*} \tan \theta \end{equation*}

Subsection A.1.2 Trigonometry — Special Triangles

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*}

Subsection 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*}

Subsection 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*}

Subsection A.1.5 Inverse Trigonometric Functions

\begin{equation*} \arcsin x \end{equation*}

\begin{equation*} \arccos x \end{equation*}

\(\arctan x\)

Domain: \(-1 \leq x \leq 1\)

Domain: all real numbers

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

Range: \(0 \leq \arccos x \leq \pi\)

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} \lt \theta \lt \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*}

\begin{equation*} \arccsc x \end{equation*}

\begin{equation*} \arcsec x \end{equation*}

\begin{equation*} \arccot x \end{equation*}

Domain: \(|x|\ge 1\)

Domain: all real numbers

Range: \(-\frac{\pi}{2} \leq \arccsc x \leq \frac{\pi}{2}\)

Range: \(0 \leq \arcsec x \leq \pi\)

Range: \(0 \lt \arccot x \lt \pi\)

\begin{equation*} \arccsc x \ne 0 \end{equation*}

\begin{equation*} \arcsec x \ne \frac{\pi}{2} \end{equation*}

Again

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

and

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

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