Trigonometry

How are $\sin\theta$, $\cos\theta$ and $\tan\theta$ defined in a right triangle?

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$.

How are $\sin\theta$ and $\cos\theta$ defined on the unit circle?

The point at angle $\theta$ (measured counterclockwise from the positive x-axis) on the unit circle is $(\cos\theta, \sin\theta)$.

How are $\csc\theta$, $\sec\theta$ and $\cot\theta$ defined?

$\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$.

How do you convert between degrees and radians?

$180^\circ = \pi$ rad, so $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$ and $\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}$.

What are $\sin$, $\cos$ and $\tan$ of $0$?

$\sin 0 = 0$, $\cos 0 = 1$, $\tan 0 = 0$.

What are $\sin$, $\cos$ and $\tan$ of $\frac{\pi}{6}$ $(30^\circ)$?

$\sin\frac{\pi}{6} = \frac{1}{2}$, $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.

What are $\sin$, $\cos$ and $\tan$ of $\frac{\pi}{4}$ $(45^\circ)$?

$\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\tan\frac{\pi}{4} = 1$.

What are $\sin$, $\cos$ and $\tan$ of $\frac{\pi}{3}$ $(60^\circ)$?

$\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{3} = \frac{1}{2}$, $\tan\frac{\pi}{3} = \sqrt{3}$.

What are $\sin$, $\cos$ and $\tan$ of $\frac{\pi}{2}$ $(90^\circ)$?

$\sin\frac{\pi}{2} = 1$, $\cos\frac{\pi}{2} = 0$, and $\tan\frac{\pi}{2}$ is undefined.

What are $\sin\pi$, $\cos\pi$, $\sin\frac{3\pi}{2}$ and $\cos\frac{3\pi}{2}$?

$\sin\pi = 0$, $\cos\pi = -1$, $\sin\frac{3\pi}{2} = -1$, $\cos\frac{3\pi}{2} = 0$.

Which trig functions are positive in each quadrant?

Quadrant I: all. Quadrant II: $\sin$ (and $\csc$). Quadrant III: $\tan$ (and $\cot$). Quadrant IV: $\cos$ (and $\sec$). Mnemonic: "All Students Take Calculus".

What is a reference angle and how is it used?

The acute angle between the terminal side of $\theta$ and the x-axis. The trig value of $\theta$ equals the value at the reference angle, with the sign set by the quadrant.

What are the periods of $\sin$, $\cos$ and $\tan$?

$\sin$ and $\cos$ have period $2\pi$, $\tan$ (and $\cot$) has period $\pi$.

Which trig functions are even and which are odd?

$\cos(-x) = \cos x$ and $\sec(-x) = \sec x$ are even. $\sin(-x) = -\sin x$, $\tan(-x) = -\tan x$, $\csc$ and $\cot$ are odd.

What are the domain and range of $\sin x$ and $\cos x$?

Domain $\mathbb{R}$, range $[-1, 1]$.

What are the domain and range of $\tan x$?

Domain: all $x \neq \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$ (vertical asymptotes there). Range: $\mathbb{R}$.

For $y = A\sin(Bx - C) + D$, what are the amplitude, period, phase shift and vertical shift?

Amplitude $\lvert A \rvert$, period $\frac{2\pi}{\lvert B \rvert}$, phase shift $\frac{C}{B}$ (to the right if positive), vertical shift $D$.

What do the graphs of $\sin x$ and $\cos x$ look like?

Both are waves between $-1$ and $1$ with period $2\pi$. $\sin x$ starts at $0$ and rises to $1$ at $\frac{\pi}{2}$. $\cos x$ starts at its maximum $1$. $\cos x = \sin\left(x + \frac{\pi}{2}\right)$, so the cosine graph is the sine graph shifted left by $\frac{\pi}{2}$.

What are the three Pythagorean identities?

$\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$.

What are the cofunction identities?

$\sin\left(\frac{\pi}{2} - x\right) = \cos x$, $\cos\left(\frac{\pi}{2} - x\right) = \sin x$, $\tan\left(\frac{\pi}{2} - x\right) = \cot x$.

What are $\sin(\pi - x)$, $\cos(\pi - x)$, $\sin(\pi + x)$ and $\cos(\pi + x)$?

$\sin(\pi - x) = \sin x$, $\cos(\pi - x) = -\cos x$, $\sin(\pi + x) = -\sin x$, $\cos(\pi + x) = -\cos x$.

What is the sum/difference formula for $\sin(a \pm b)$?

$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$.

What is the sum/difference formula for $\cos(a \pm b)$?

$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$ (note the sign flips).

What is the sum/difference formula for $\tan(a \pm b)$?

$\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$.

What is the double-angle formula for $\sin 2x$?

$\sin 2x = 2\sin x \cos x$.

What are the three forms of the double-angle formula for $\cos 2x$?

$\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$.

What is the double-angle formula for $\tan 2x$?

$\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$.

What are the power-reduction formulas for $\sin^2 x$ and $\cos^2 x$?

$\sin^2 x = \frac{1 - \cos 2x}{2}$, $\cos^2 x = \frac{1 + \cos 2x}{2}$. Essential for integrating even powers of $\sin$ and $\cos$.

What are the half-angle formulas for $\sin\frac{x}{2}$ and $\cos\frac{x}{2}$?

$\sin\frac{x}{2} = \pm\sqrt{\frac{1 - \cos x}{2}}$, $\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$, with the sign chosen by the quadrant of $\frac{x}{2}$.

What is the half-angle formula for $\tan\frac{x}{2}$ (sign-free forms)?

$\tan\frac{x}{2} = \frac{\sin x}{1 + \cos x} = \frac{1 - \cos x}{\sin x}$.

What are the triple-angle formulas for $\sin 3x$ and $\cos 3x$?

$\sin 3x = 3\sin x - 4\sin^3 x$, $\cos 3x = 4\cos^3 x - 3\cos x$.

What are the product-to-sum formulas?

$\sin a \cos b = \frac{1}{2}[\sin(a+b) + \sin(a-b)]$, $\cos a \cos b = \frac{1}{2}[\cos(a-b) + \cos(a+b)]$, $\sin a \sin b = \frac{1}{2}[\cos(a-b) - \cos(a+b)]$.

What are the sum-to-product formulas for $\sin a \pm \sin b$?

$\sin a + \sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2}$, $\sin a - \sin b = 2\cos\frac{a+b}{2}\sin\frac{a-b}{2}$.

What are the sum-to-product formulas for $\cos a \pm \cos b$?

$\cos a + \cos b = 2\cos\frac{a+b}{2}\cos\frac{a-b}{2}$, $\cos a - \cos b = -2\sin\frac{a+b}{2}\sin\frac{a-b}{2}$.

How do you write $a\sin x + b\cos x$ as a single sine (harmonic addition)?

$a\sin x + b\cos x = R\sin(x + \varphi)$, where $R = \sqrt{a^2 + b^2}$, $\cos\varphi = \frac{a}{R}$, $\sin\varphi = \frac{b}{R}$. Pick the quadrant of $\varphi$ from the signs of both $\cos\varphi$ and $\sin\varphi$, not from $\tan\varphi = \frac{b}{a}$ alone.

What is the general solution of $\sin x = c$ for $-1 \le c \le 1$?

$x = \arcsin c + 2k\pi$ or $x = \pi - \arcsin c + 2k\pi$, $k \in \mathbb{Z}$, where $\arcsin c \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is the principal value.

What is the general solution of $\cos x = c$ for $-1 \le c \le 1$?

$x = \pm\arccos c + 2k\pi$, $k \in \mathbb{Z}$, where $\arccos c \in [0, \pi]$ is the principal value.

What is the general solution of $\tan x = c$?

$x = \arctan c + k\pi$, $k \in \mathbb{Z}$, where $\arctan c \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ is the principal value.

What is the Law of Sines?

In any triangle with sides $a, b, c$ opposite angles $A, B, C$: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$, where $R$ is the circumradius.

What is the Law of Cosines?

$c^2 = a^2 + b^2 - 2ab\cos C$ (and similarly for the other sides). It reduces to Pythagoras when $C = \frac{\pi}{2}$.

What is the area of a triangle in terms of two sides and the included angle?

$\text{Area} = \frac{1}{2}ab\sin C$.

What are the domain and range of $\arcsin x$?

Domain $[-1, 1]$, range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

What are the domain and range of $\arccos x$?

Domain $[-1, 1]$, range $[0, \pi]$.

What are the domain and range of $\arctan x$?

Domain $\mathbb{R}$, range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ (horizontal asymptotes $y = \pm\frac{\pi}{2}$).

What is $\arcsin x + \arccos x$?

$\arcsin x + \arccos x = \frac{\pi}{2}$ for all $x \in [-1, 1]$.

What is $\cos(\arcsin x)$ and $\sin(\arccos x)$?

Both equal $\sqrt{1 - x^2}$, the nonnegative root, because $\arcsin x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ where $\cos \ge 0$, and $\arccos x \in [0, \pi]$ where $\sin \ge 0$.

What is $\sec(\arctan x)$ and $\sin(\arctan x)$?

$\sec(\arctan x) = \sqrt{1 + x^2}$, $\sin(\arctan x) = \frac{x}{\sqrt{1 + x^2}}$.

What are the key features of the graphs of $\arcsin x$, $\arccos x$ and $\arctan x$?

$\arcsin$: increasing, odd, from $\left(-1, -\frac{\pi}{2}\right)$ to $\left(1, \frac{\pi}{2}\right)$ through the origin. $\arccos$: decreasing, from $(-1, \pi)$ to $(1, 0)$ through $\left(0, \frac{\pi}{2}\right)$. $\arctan$: increasing, odd, horizontal asymptotes $y = \pm\frac{\pi}{2}$. Each is the reflection of the restricted original function in $y = x$.

How do you solve $\sin(kx) = c$ on an interval such as $[0, 2\pi)$?

Let $u = kx$, so $u$ runs over $[0, 2k\pi)$. Find every solution for $u$ in that wider interval, then divide by $k$. Expect up to $2k$ solutions, not just 2.

How do you solve an equation like $2\sin^2 x - \sin x - 1 = 0$?

Substitute $t = \sin x$: $2t^2 - t - 1 = 0$ gives $t = 1$ or $t = -\frac{1}{2}$. Discard any root outside $[-1, 1]$, then solve each: $x = \frac{\pi}{2} + 2k\pi$, $x = -\frac{\pi}{6} + 2k\pi$, $x = \frac{7\pi}{6} + 2k\pi$.

How do you solve an equation that mixes $\cos^2 x$ and $\sin x$, such as $2\cos^2 x + 3\sin x = 3$?

Use $\cos^2 x = 1 - \sin^2 x$ to write it in one function: $2\sin^2 x - 3\sin x + 1 = 0$, so $\sin x = 1$ or $\sin x = \frac{1}{2}$. Solutions: $x = \frac{\pi}{2} + 2k\pi$, $x = \frac{\pi}{6} + 2k\pi$, $x = \frac{5\pi}{6} + 2k\pi$.

Why should you not divide both sides of a trig equation by $\sin x$ or $\cos x$?

You lose the solutions where that factor is zero. Factor instead: $\sin x\cos x = \sin x \Rightarrow \sin x(\cos x - 1) = 0$, so $\sin x = 0$ or $\cos x = 1$.

What is $\lim_{x \to 0} \frac{\sin x}{x}$?

$\lim_{x \to 0} \frac{\sin x}{x} = 1$ ($x$ in radians).

What is $\lim_{x \to 0} \frac{1 - \cos x}{x}$ and $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$?

$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ and $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$.

What are the derivatives of $\sin x$, $\cos x$ and $\tan x$?

$(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x = 1 + \tan^2 x$.

What are the derivatives of $\csc x$, $\sec x$ and $\cot x$?

$(\csc x)' = -\csc x \cot x$, $(\sec x)' = \sec x \tan x$, $(\cot x)' = -\csc^2 x$.

What are the derivatives of $\arcsin x$, $\arccos x$ and $\arctan x$?

$(\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}$, $(\arccos x)' = -\frac{1}{\sqrt{1 - x^2}}$, $(\arctan x)' = \frac{1}{1 + x^2}$.

What are $\int \sin x \, dx$ and $\int \cos x \, dx$?

$\int \sin x \, dx = -\cos x + C$, $\int \cos x \, dx = \sin x + C$.

What are $\int \tan x \, dx$ and $\int \cot x \, dx$?

$\int \tan x \, dx = -\ln\lvert\cos x\rvert + C$, $\int \cot x \, dx = \ln\lvert\sin x\rvert + C$.

What are $\int \sec x \, dx$ and $\int \csc x \, dx$?

$\int \sec x \, dx = \ln\lvert\sec x + \tan x\rvert + C$, $\int \csc x \, dx = -\ln\lvert\csc x + \cot x\rvert + C$.

What are $\int \sin^2 x \, dx$ and $\int \cos^2 x \, dx$?

$\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$, $\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$.

What are $\int \frac{dx}{\sqrt{a^2 - x^2}}$ and $\int \frac{dx}{a^2 + x^2}$?

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C$, $\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C$.

Which trig substitution do you use for $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$ and $\sqrt{x^2 - a^2}$?

$\sqrt{a^2 - x^2}$: $x = a\sin\theta$. $\sqrt{a^2 + x^2}$: $x = a\tan\theta$. $\sqrt{x^2 - a^2}$: $x = a\sec\theta$.

What is the Weierstrass substitution $t = \tan\frac{x}{2}$?

$\sin x = \frac{2t}{1 + t^2}$, $\cos x = \frac{1 - t^2}{1 + t^2}$, $dx = \frac{2\,dt}{1 + t^2}$. It turns rational functions of $\sin$ and $\cos$ into rational functions of $t$.

What are the derivatives of $\sin u$, $\cos u$ and $\tan u$ where $u = u(x)$ (chain rule)?

$\frac{d}{dx}\sin u = \cos u \cdot u'$, $\frac{d}{dx}\cos u = -\sin u \cdot u'$, $\frac{d}{dx}\tan u = \sec^2 u \cdot u'$.

What are the derivatives of $\arcsin u$, $\arccos u$ and $\arctan u$ where $u = u(x)$ (chain rule)?

$\frac{u'}{\sqrt{1 - u^2}}$, $-\frac{u'}{\sqrt{1 - u^2}}$, $\frac{u'}{1 + u^2}$.

How do you differentiate $\sin y$ and $\cos(xy)$ implicitly with respect to $x$?

Treat $y$ as a function of $x$ and use the chain rule: $\frac{d}{dx}\sin y = \cos y \cdot \frac{dy}{dx}$, $\frac{d}{dx}\cos(xy) = -\sin(xy)\left(y + x\frac{dy}{dx}\right)$.

How do you integrate $\int \sin^m x\cos^n x\,dx$?

Odd power of $\sin$: save one $\sin x$, convert the rest with $\sin^2 x = 1 - \cos^2 x$, let $u = \cos x$. Odd power of $\cos$: save one $\cos x$, convert the rest with $\cos^2 x = 1 - \sin^2 x$, let $u = \sin x$. Both even: use the power-reduction formulas.

How do you integrate $\int \tan^m x\sec^n x\,dx$?

Even power of $\sec$: save $\sec^2 x$, convert the rest with $\sec^2 x = 1 + \tan^2 x$, let $u = \tan x$. Odd power of $\tan$: save $\sec x\tan x$, convert the rest with $\tan^2 x = \sec^2 x - 1$, let $u = \sec x$.

What is $\int \frac{du}{u\sqrt{u^2 - a^2}}$?

$\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a}\,\text{arcsec}\frac{\lvert u \rvert}{a} + C$.

How do you integrate $\int \frac{dx}{x^2 + 2x + 5}$?

Complete the square: $x^2 + 2x + 5 = (x + 1)^2 + 4$. With $u = x + 1$ this is $\int \frac{du}{u^2 + 4} = \frac{1}{2}\arctan\frac{x + 1}{2} + C$.

What is $\lim_{x \to 0} \frac{\sin(ax)}{bx}$?

$\frac{a}{b}$. Rewrite it as $\frac{a}{b} \cdot \frac{\sin(ax)}{ax}$ and use $\lim_{t \to 0} \frac{\sin t}{t} = 1$.

Which technique fits $\int \frac{x}{1 + x^4}\,dx$?

Substitute $u = x^2$, $du = 2x\,dx$: $\frac{1}{2}\int \frac{du}{1 + u^2} = \frac{1}{2}\arctan(x^2) + C$.

Which technique fits $\int \sin^3 x\cos^2 x\,dx$?

Odd power of $\sin$: write $\sin^3 x = (1 - \cos^2 x)\sin x$ and let $u = \cos x$: $-\int (1 - u^2)u^2\,du = -\frac{\cos^3 x}{3} + \frac{\cos^5 x}{5} + C$.

Which technique fits $\int \sqrt{9 - x^2}\,dx$?

Trig substitution $x = 3\sin\theta$, $dx = 3\cos\theta\,d\theta$, $\sqrt{9 - x^2} = 3\cos\theta$, giving $9\int \cos^2\theta\,d\theta$. Power reduction then gives $\frac{9}{2}\arcsin\frac{x}{3} + \frac{x}{2}\sqrt{9 - x^2} + C$.

What are the Taylor series of $\sin x$ and $\cos x$ about $0$?

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$.

What are the small-angle approximations?

For small $x$ (radians): $\sin x \approx x$, $\tan x \approx x$, $\cos x \approx 1 - \frac{x^2}{2}$.

What is Euler's formula?

$e^{i\theta} = \cos\theta + i\sin\theta$.

How are $\cos\theta$ and $\sin\theta$ expressed with complex exponentials?

$\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$, $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$.

What is the polar (trigonometric) form of a complex number $z = a + bi$?

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, where $r = \lvert z \rvert = \sqrt{a^2 + b^2}$ and $\theta = \arg z$ with $\tan\theta = \frac{b}{a}$ (quadrant chosen by the signs of $a, b$).

What is de Moivre's theorem?

$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ for integer $n$, equivalently $(re^{i\theta})^n = r^n e^{in\theta}$.

How do you find the $n$-th roots of $z = re^{i\theta}$?

$z_k = r^{1/n} e^{i\frac{\theta + 2k\pi}{n}}$ for $k = 0, 1, \ldots, n-1$. They are equally spaced on a circle of radius $r^{1/n}$.

What is the 2D rotation matrix for a counterclockwise rotation by $\theta$?

$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, with $\det R(\theta) = 1$ and $R(\theta)^{-1} = R(\theta)^T = R(-\theta)$.

How is the angle between two vectors $\mathbf{u}$ and $\mathbf{v}$ found?

$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\lVert\mathbf{u}\rVert\,\lVert\mathbf{v}\rVert}$, and in $\mathbb{R}^3$ also $\lVert\mathbf{u} \times \mathbf{v}\rVert = \lVert\mathbf{u}\rVert\,\lVert\mathbf{v}\rVert\sin\theta$.

How are Cartesian and polar coordinates related?

$x = r\cos\theta$, $y = r\sin\theta$, $r = \sqrt{x^2 + y^2}$, $\tan\theta = \frac{y}{x}$. Choose the quadrant of $\theta$ from the signs of $x$ and $y$.

What are the slope and area formulas for a polar curve $r = f(\theta)$?

Slope: $\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$. Area swept from $\alpha$ to $\beta$: $A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$.

How do you differentiate a parametric curve $x = x(t)$, $y = y(t)$?

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$. Example: for $x = \cos t$, $y = \sin t$, $\frac{dy}{dx} = -\cot t$.

How are $\sinh x$ and $\cosh x$ defined, and what is their main identity?

$\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$, $\tanh x = \frac{\sinh x}{\cosh x}$, and $\cosh^2 x - \sinh^2 x = 1$ (note the minus sign, unlike $\cos^2 + \sin^2$).

What are the derivatives of $\sinh x$, $\cosh x$ and $\tanh x$?

$(\sinh x)' = \cosh x$, $(\cosh x)' = \sinh x$ (no minus sign), $(\tanh x)' = \text{sech}^2 x = 1 - \tanh^2 x$.