Precalc
← Back to Exam Overview

📋 Exam 3 — Formula Sheet

All key formulas from 11 sections at a glance.

§5.1

Angles & Radian Measure
Degrees to Radians
θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Multiply the degree measure by π/180

Radians to Degrees
θdeg=θrad×180π\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}

Multiply the radian measure by 180/π

Arc Length
s=rθs = r\theta

Where s is arc length, r is radius, and θ is the angle in radians

Area of a Sector
A=12r2θA = \frac{1}{2}r^2\theta

Only works when θ is in radians

Coterminal Angles
θ±360°norθ±2πn(n=1,2,3,…)\theta \pm 360°n \quad \text{or} \quad \theta \pm 2\pi n \quad (n = 1, 2, 3, \ldots)

Add or subtract full rotations to find angles that land in the same position

§5.2

Unit Circle — Sine & Cosine
Unit circle definitions
cos⁡θ=x,sin⁡θ=y\cos\theta = x, \quad \sin\theta = y

where (x,y)(x, y) is the point on the unit circle at angle θ\theta

Pythagorean identity
cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1

follows directly from x2+y2=1x^2 + y^2 = 1

Key first-quadrant values
sin⁡0=0,  sin⁡π6=12,  sin⁡π4=22,  sin⁡π3=32,  sin⁡π2=1\sin 0 = 0,\; \sin\frac{\pi}{6} = \frac{1}{2},\; \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2},\; \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2},\; \sin\frac{\pi}{2} = 1
Reference angle
ref(θ)=acute angle between terminal side and x-axis\text{ref}(\theta) = \text{acute angle between terminal side and } x\text{-axis}

use the reference angle to find trig values in any quadrant

Even/odd identities
cos⁡(−θ)=cos⁡θ,sin⁡(−θ)=−sin⁡θ\cos(-\theta) = \cos\theta, \quad \sin(-\theta) = -\sin\theta

cosine is even, sine is odd

§5.3

The Other Trigonometric Functions
Tangent and cotangent
tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}
Secant and cosecant
sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}
Reciprocal pairs (product form)
tan⁡θ⋅cot⁡θ=1,sec⁡θ⋅cos⁡θ=1,csc⁡θ⋅sin⁡θ=1\tan\theta \cdot \cot\theta = 1, \quad \sec\theta \cdot \cos\theta = 1, \quad \csc\theta \cdot \sin\theta = 1

Each pair multiplies to 1 — useful for simplifying expressions

Pythagorean identity (tangent form)
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

divide cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1 by cos⁡2θ\cos^2\theta

Pythagorean identity (cotangent form)
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

divide cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1 by sin⁡2θ\sin^2\theta

§5.4

Right Triangle Trigonometry
SOH-CAH-TOA
sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}
Cofunction identity (sine–cosine)
sin⁡θ=cos⁡ ⁣(π2−θ),cos⁡θ=sin⁡ ⁣(π2−θ)\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right), \quad \cos\theta = \sin\!\left(\frac{\pi}{2} - \theta\right)
Cofunction identity (tangent–cotangent)
tan⁡θ=cot⁡ ⁣(π2−θ),cot⁡θ=tan⁡ ⁣(π2−θ)\tan\theta = \cot\!\left(\frac{\pi}{2} - \theta\right), \quad \cot\theta = \tan\!\left(\frac{\pi}{2} - \theta\right)
Cofunction identity (secant–cosecant)
sec⁡θ=csc⁡ ⁣(π2−θ),csc⁡θ=sec⁡ ⁣(π2−θ)\sec\theta = \csc\!\left(\frac{\pi}{2} - \theta\right), \quad \csc\theta = \sec\!\left(\frac{\pi}{2} - \theta\right)
Pythagorean theorem
a2+b2=c2a^2 + b^2 = c^2

where cc is the hypotenuse

§6.1

Graphs of Sine & Cosine
General Sinusoidal Form
y=Asin⁡(Bx−C)+Dory=Acos⁡(Bx−C)+Dy = A\sin(Bx - C) + D \quad \text{or} \quad y = A\cos(Bx - C) + D

A, B, C, D are real constants with B > 0

Amplitude
Amplitude=∣A∣\text{Amplitude} = |A|

The height from the midline to a peak. If A is negative, the graph is reflected vertically.

Period
Period=2π∣B∣\text{Period} = \frac{2\pi}{|B|}

The horizontal length of one full cycle

Phase Shift
Phase Shift=CB\text{Phase Shift} = \frac{C}{B}

Positive means shift right, negative means shift left

Vertical Shift
Vertical Shift=D\text{Vertical Shift} = D

The midline of the graph is y = D

§6.2

Graphs of Other Trig Functions
Period of Tangent / Cotangent
Period=π∣B∣\text{Period} = \frac{\pi}{|B|}

Half the period of sine and cosine for the same B value

Period of Secant / Cosecant
Period=2π∣B∣\text{Period} = \frac{2\pi}{|B|}

Same period formula as sine and cosine

Tangent Asymptotes (standard)
x=π2+nπ,n∈Zx = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}

Where cos(x) = 0

Cotangent Asymptotes (standard)
x=nπ,n∈Zx = n\pi, \quad n \in \mathbb{Z}

Where sin(x) = 0

General Tangent Form
y=Atan⁡(Bx−C)+Dy = A\tan(Bx - C) + D

Phase shift = C/B, vertical stretch = |A|, vertical shift = D

§6.3

Inverse Trigonometric Functions
Arcsine
y=arcsin⁡(x):Domain [−1,1],Range [−π2,π2]y = \arcsin(x): \quad \text{Domain } [-1, 1], \quad \text{Range } \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
Arccosine
y=arccos⁡(x):Domain [−1,1],Range [0,π]y = \arccos(x): \quad \text{Domain } [-1, 1], \quad \text{Range } [0, \pi]
Arctangent
y=arctan⁡(x):Domain (−∞,∞),Range (−π2,π2)y = \arctan(x): \quad \text{Domain } (-\infty, \infty), \quad \text{Range } \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
Cancellation (inner inverse)
sin⁡(arcsin⁡(x))=x   for x∈[−1,1];arcsin⁡(sin⁡(x))=x   for x∈[−π2,π2]\sin(\arcsin(x)) = x \;\text{ for } x \in [-1,1]; \quad \arcsin(\sin(x)) = x \;\text{ for } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

Same pattern holds for cos/arccos and tan/arctan with their respective domains

Composition via Triangle
sin⁡(arccos⁡(x))=1−x2,cos⁡(arcsin⁡(x))=1−x2\sin(\arccos(x)) = \sqrt{1 - x^2}, \quad \cos(\arcsin(x)) = \sqrt{1 - x^2}

Derived from a right triangle with hypotenuse 1

§7.1

Trig Identities
Pythagorean Identity
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

The most important identity. Rearranges to give you sin² or cos² alone.

Pythagorean Identity (tan/sec)
1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

Divide the main Pythagorean identity by cos²(θ) to get this one.

Pythagorean Identity (cot/csc)
1+cot⁡2(θ)=csc⁡2(θ)1 + \cot^2(\theta) = \csc^2(\theta)

Divide the main Pythagorean identity by sin²(θ) to get this one.

Quotient Identities
tan⁡(θ)=sin⁡(θ)cos⁡(θ),cot⁡(θ)=cos⁡(θ)sin⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
Reciprocal Identities
csc⁡(θ)=1sin⁡(θ),sec⁡(θ)=1cos⁡(θ),cot⁡(θ)=1tan⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)}

§7.2

Sum & Difference Identities
Sine of a Sum
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B
Sine of a Difference
sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B
Cosine of a Sum
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B

Watch the sign — it's minus for the sum, which is the opposite of what you might guess.

Cosine of a Difference
cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B
Tangent of a Sum/Difference
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

The sign in the denominator is opposite to the sign in the numerator.

§7.3

Double-Angle & Half-Angle Formulas
Sine Double-Angle
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
Cosine Double-Angle (3 forms)
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Choose the form that best fits what you already know (sin only, cos only, or both).

Tangent Double-Angle
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
Half-Angle: Sine
sin⁡α2=±1−cos⁡α2\sin\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{2}}

The ± depends on the quadrant of α/2.

Half-Angle: Cosine
cos⁡α2=±1+cos⁡α2\cos\frac{\alpha}{2} = \pm\sqrt{\frac{1 + \cos\alpha}{2}}

The ± depends on the quadrant of α/2.

Power-Reducing: Sine
sin⁡2θ=1−cos⁡(2θ)2\sin^2\theta = \frac{1 - \cos(2\theta)}{2}

Derived from the cosine double-angle formula; replaces a square with a first-power expression.

Power-Reducing: Cosine
cos⁡2θ=1+cos⁡(2θ)2\cos^2\theta = \frac{1 + \cos(2\theta)}{2}

Derived from the cosine double-angle formula; replaces a square with a first-power expression.

Half-Angle: Tangent
tan⁡α2=1−cos⁡αsin⁡α=sin⁡α1+cos⁡α\tan\frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 + \cos\alpha}

Two equivalent forms — pick whichever avoids a zero denominator.

§7.5

Solving Trigonometric Equations
General Solution (Sine/Cosine)
x=x0+2nπ,n∈Zx = x_0 + 2n\pi, \quad n \in \mathbb{Z}

Sine and cosine repeat every 2π.

General Solution (Tangent)
x=x0+nπ,n∈Zx = x_0 + n\pi, \quad n \in \mathbb{Z}

Tangent repeats every π.

Zero Product Property
AB=0  ⟹  A=0 or B=0AB = 0 \implies A = 0 \text{ or } B = 0

Factor and set each factor to zero — never divide by a trig expression.

Quadratic Substitution
asin⁡2(x)+bsin⁡(x)+c=0  ⟹  let u=sin⁡(x)a\sin^2(x) + b\sin(x) + c = 0 \implies \text{let } u = \sin(x)

Treat it like au² + bu + c = 0, solve for u, then find x.