Trigonometry – Complete Reference Guide

Angle Measurement

Degree to Radian

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180°}$

$\text{Degrees} = \text{Radians} \times \frac{180°}{\pi}$

Common Conversions

$90° = \pi/2$ rad

$180° = \pi$ rad

$360° = 2\pi$ rad

Arc Length & Area

Arc: $s = r\theta$ (θ in radians)

Sector: $A = \frac{1}{2}r^2\theta$

Right Triangle Trig

SOH CAH TOA

$\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$

$\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$

$\tan\theta = \frac{\text{Opp}}{\text{Adj}}$

Reciprocal Functions

$\csc\theta = \frac{1}{\sin\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta}$

Use for angles in right triangles (0° to 90°)

Unit Circle

Key Angles Table

°	rad	sin	cos
0	0	0	1
30	π/6	1/2	√3/2
45	π/4	√2/2	√2/2
60	π/3	√3/2	1/2
90	π/2	1	0

$(x,y) = (\cos\theta, \sin\theta)$

ASTC: All-Sin-Tan-Cos

Trig Functions

Primary Functions

$\sin\theta = y/r$

$\cos\theta = x/r$

$\tan\theta = y/x$

Reciprocal Functions

$\csc\theta = r/y$

$\sec\theta = r/x$

$\cot\theta = x/y$

Quotient Form

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

$\cot\theta = \frac{\cos\theta}{\sin\theta}$

Graphs & Amplitude

General Form

$y = A\sin(B(x-C)) + D$

Parameters

$A$: Amplitude = $|A|$

$B$: Period = $2\pi/|B|$

$C$: Phase Shift (right)

$D$: Vertical Shift

Standard Periods

sin, cos: $2\pi$

tan, cot: $\pi$

sec, csc: $2\pi$

Pythagorean Identities

Fundamental

$\sin^2\theta + \cos^2\theta = 1$

Derived from Above

$\sin^2\theta = 1 - \cos^2\theta$

$\cos^2\theta = 1 - \sin^2\theta$

Tangent Form

$\tan^2\theta + 1 = \sec^2\theta$

$1 + \cot^2\theta = \csc^2\theta$

Use to simplify and prove identities

Reciprocal & Quotient

Reciprocal Identities

$\csc\theta = \frac{1}{\sin\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta}$

Quotient Identities

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

$\cot\theta = \frac{\cos\theta}{\sin\theta}$

Use to convert between functions

Cofunction Identities

Complementary Angles

$\sin(\frac{\pi}{2} - \theta) = \cos\theta$

$\cos(\frac{\pi}{2} - \theta) = \sin\theta$

$\tan(\frac{\pi}{2} - \theta) = \cot\theta$

$\cot(\frac{\pi}{2} - \theta) = \tan\theta$

$\sec(\frac{\pi}{2} - \theta) = \csc\theta$

$\csc(\frac{\pi}{2} - \theta) = \sec\theta$

Complementary angles sum to 90°

Even/Odd Identities

Odd Functions

$\sin(-\theta) = -\sin\theta$

$\tan(-\theta) = -\tan\theta$

$\csc(-\theta) = -\csc\theta$

$\cot(-\theta) = -\cot\theta$

Even Functions

$\cos(-\theta) = \cos\theta$

$\sec(-\theta) = \sec\theta$

Odd: $f(-x) = -f(x)$; Even: $f(-x) = f(x)$

Sum & Difference

Sine

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

Cosine

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

Tangent

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Note: Use ∓ (opposite) in cos formula

Double Angle

Sine

$\sin(2\theta) = 2\sin\theta\cos\theta$

Cosine (3 forms)

$\cos(2\theta) = \cos^2\theta - \sin^2\theta$

$\cos(2\theta) = 2\cos^2\theta - 1$

$\cos(2\theta) = 1 - 2\sin^2\theta$

Tangent

$\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$

Half Angle

Sine & Cosine

$\sin(\theta/2) = \pm\sqrt{\frac{1-\cos\theta}{2}}$

$\cos(\theta/2) = \pm\sqrt{\frac{1+\cos\theta}{2}}$

Tangent (2 forms)

$\tan(\theta/2) = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$

$\tan(\theta/2) = \frac{\sin\theta}{1+\cos\theta}$

Sign depends on quadrant

Product-to-Sum

$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$

$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$

Sum-to-Product

$\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$

$\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$

Inverse Functions

Domains & Ranges

$\sin^{-1}(x)$: Domain $[-1,1]$

Range: $[-\pi/2, \pi/2]$

$\cos^{-1}(x)$: Domain $[-1,1]$

Range: $[0, \pi]$

$\tan^{-1}(x)$: Domain $\mathbb{R}$

Range: $(-\pi/2, \pi/2)$

Relationships

$\sin^{-1}(x) + \cos^{-1}(x) = \pi/2$

Solving Equations

General Solutions

$\sin\theta = a$: $\theta = \sin^{-1}(a) + 2\pi k$ or $\pi - \sin^{-1}(a) + 2\pi k$

$\cos\theta = a$: $\theta = \pm\cos^{-1}(a) + 2\pi k$

$\tan\theta = a$: $\theta = \tan^{-1}(a) + \pi k$

Strategy

Isolate trig function

Use inverse to find reference angle

Add periodic solutions

k ∈ integers

Law of Sines

Formula

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

When to Use

AAS (Angle-Angle-Side)

ASA (Angle-Side-Angle)

SSA (Side-Side-Angle)

Ambiguous Case (SSA)

No solution if $a < b\sin A$

One solution if $a = b\sin A$

Two solutions if $b\sin A < a < b$

For non-right triangles

Law of Cosines

Formulas

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

Finding Angles

$\cos A = \frac{b^2+c^2-a^2}{2bc}$

When to Use

SAS (Side-Angle-Side)

SSS (Side-Side-Side)

Extension of Pythagorean theorem

Triangle Area

SAS Formula

$A = \frac{1}{2}ab\sin C$

$A = \frac{1}{2}bc\sin A$

$A = \frac{1}{2}ac\sin B$

Heron's Formula (SSS)

$s = \frac{a+b+c}{2}$ (semiperimeter)

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Right Triangle

$A = \frac{1}{2}ab$ (legs)

Polar Coordinates

Conversion Formulas

$x = r\cos\theta$

$y = r\sin\theta$

$r^2 = x^2 + y^2$

$\tan\theta = y/x$

Key Facts

$r \geq 0$ (distance from origin)

$\theta$ in radians or degrees

Multiple representations: $(r,\theta) = (-r, \theta+\pi)$

Useful for circular and spiral patterns

Complex & De Moivre

Polar Form

$z = a + bi = r(\cos\theta + i\sin\theta)$

$r = |z| = \sqrt{a^2+b^2}$

$\theta = \arg(z) = \arctan(b/a)$

De Moivre's Theorem

$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$

nth Roots

$z^{1/n} = r^{1/n}(\cos(\frac{\theta+2\pi k}{n}) + i\sin(\frac{\theta+2\pi k}{n}))$

$k = 0, 1, 2, ..., n-1$

TRIGONOMETRY Complete Reference