Trigonometry

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Section A.1 Trigonometry

Subsection A.1.1 Trigonometry — Graphs

\sin θ

\cos θ

\tan θ

Subsection A.1.2 Trigonometry — Special Triangles

From the above pair of special triangles we have

\begin{aligned} \sin \frac{π}{4} & = \frac{1}{\sqrt{2}} & \sin \frac{π}{6} & = \frac{1}{2} & \sin \frac{π}{3} & = \frac{\sqrt{3}}{2} \\ \cos \frac{π}{4} & = \frac{1}{\sqrt{2}} & \cos \frac{π}{6} & = \frac{\sqrt{3}}{2} & \cos \frac{π}{3} & = \frac{1}{2} \\ \tan \frac{π}{4} & = 1 & \tan \frac{π}{6} & = \frac{1}{\sqrt{3}} & \tan \frac{π}{3} & = \sqrt{3} \end{aligned}

Subsection A.1.3 Trigonometry — Simple Identities

Periodicity
$\begin{aligned} \sin (θ + 2 π) & = \sin (θ) & \cos (θ + 2 π) & = \cos (θ) \end{aligned}$
Reflection
$\begin{aligned} \sin (- θ) & = - \sin (θ) & \cos (- θ) & = \cos (θ) \end{aligned}$
Reflection around $π / 4$
$\begin{aligned} \sin (\frac{π}{2} - θ) & = \cos θ & \cos (\frac{π}{2} - θ) & = \sin θ \end{aligned}$
Reflection around $π / 2$
$\begin{aligned} \sin (π - θ) & = \sin θ & \cos (π - θ) & = - \cos θ \end{aligned}$
Rotation by $π$
$\begin{aligned} \sin (θ + π) & = - \sin θ & \cos (θ + π) & = - \cos θ \end{aligned}$
Pythagoras
$\begin{aligned} \sin^{2} θ + \cos^{2} θ & = 1 \\ \tan^{2} θ + 1 & = \sec^{2} θ \\ 1 + \cot^{2} θ & = \csc^{2} θ \end{aligned}$
$\sin$ and $\cos$ building blocks
$\begin{matrix} \tan θ = \frac{\sin θ}{\cos θ} \csc θ = \frac{1}{\sin θ} \sec θ = \frac{1}{\cos θ} \cot θ = \frac{\cos θ}{\sin θ} = \frac{1}{\tan θ} \end{matrix}$

Subsection A.1.4 Trigonometry — Add and Subtract Angles

Sine
$\begin{aligned} \sin (α \pm β) & = \sin (α) \cos (β) \pm \cos (α) \sin (β) \end{aligned}$
Cosine
$\begin{aligned} \cos (α \pm β) & = \cos (α) \cos (β) \mp \sin (α) \sin (β) \end{aligned}$
Tangent
$\begin{aligned} \tan (α + β) & = \frac{\tan α + \tan β}{1 - \tan α \tan β} \\ \tan (α - β) & = \frac{\tan α - \tan β}{1 + \tan α \tan β} \end{aligned}$
Double angle
$\begin{aligned} \sin (2 θ) & = 2 \sin (θ) \cos (θ) \\ \cos (2 θ) & = \cos^{2} (θ) - \sin^{2} (θ) \\ = 2 \cos^{2} (θ) - 1 \\ = 1 - 2 \sin^{2} (θ) \\ \tan (2 θ) & = \frac{2 \tan (θ)}{1 - \tan^{2} θ} \\ \cos^{2} θ & = \frac{1 + \cos (2 θ)}{2} \\ \sin^{2} θ & = \frac{1 - \cos (2 θ)}{2} \\ \tan^{2} θ & = \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{aligned}$
Products to sums
$\begin{aligned} \sin (α) \cos (β) & = \frac{\sin (α + β) + \sin (α - β)}{2} \\ \sin (α) \sin (β) & = \frac{\cos (α - β) - \cos (α + β)}{2} \\ \cos (α) \cos (β) & = \frac{\cos (α - β) + \cos (α + β)}{2} \end{aligned}$
Sums to products
$\begin{aligned} \sin α + \sin β & = 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2} \\ \sin α - \sin β & = 2 \cos \frac{α + β}{2} \sin \frac{α - β}{2} \\ \cos α + \cos β & = 2 \cos \frac{α + β}{2} \cos \frac{α - β}{2} \\ \cos α - \cos β & = - 2 \sin \frac{α + β}{2} \sin \frac{α - β}{2} \end{aligned}$

Subsection A.1.5 Inverse Trigonometric Functions

\arcsin x

\arccos x

\arctan x

Domain:

- 1 \leq x \leq 1

Domain:

- 1 \leq x \leq 1

Domain: all real numbers

Range:

- \frac{π}{2} \leq \arcsin x \leq \frac{π}{2}

Range:

0 \leq \arccos x \leq π

Range:

- \frac{π}{2} < \arctan x < \frac{π}{2}

Since these functions are inverses of each other we have

\begin{aligned} \arcsin (\sin θ) & = θ & - \frac{π}{2} \leq θ \leq \frac{π}{2} \\ \arccos (\cos θ) & = θ & 0 \leq θ \leq π \\ \arctan (\tan θ) & = θ & - \frac{π}{2} < θ < \frac{π}{2} \end{aligned}

and also

\begin{aligned} \sin (\arcsin x) & = x & - 1 \leq x \leq 1 \\ \cos (\arccos x) & = x & - 1 \leq x \leq 1 \\ \tan (\arctan x) & = x & any real x \end{aligned}

arccsc x

arcsec x

arccot x

Domain:

| x | \geq 1

Domain:

| x | \geq 1

Domain: all real numbers

Range:

- \frac{π}{2} \leq arccsc x \leq \frac{π}{2}

Range:

0 \leq arcsec x \leq π

Range:

0 < arccot x < π

arccsc x \neq 0

arcsec x \neq \frac{π}{2}

Again

\begin{aligned} arccsc (\csc θ) & = θ & - \frac{π}{2} \leq θ \leq \frac{π}{2}, θ \neq 0 \\ arcsec (\sec θ) & = θ & 0 \leq θ \leq π, θ \neq \frac{π}{2} \\ arccot (\cot θ) & = θ & 0 < θ < π \end{aligned}

and

\begin{aligned} \csc (arccsc x) & = x & | x | \geq 1 \\ \sec (arcsec x) & = x & | x | \geq 1 \\ \cot (arccot x) & = x & any real x \end{aligned}