A Quick Review Of Trigonometry

Trigonometric functions model periodic phenomena and geometric relationships. Mastering radian measures, fundamental identities, double-angle formulas, and triangle laws is indispensable for calculus.

Quick Reference

Category	Primary Identity / Formula	Description
Radian Conversion	$180^{\circ} = π radians$	Angle measurement conversion
Arc Length & Sector	$s = r θ, A = \frac{1}{2} r^{2} θ$	For central angle $θ$ in radians
Pythagorean Identity	$\sin^{2} θ + \cos^{2} θ = 1$	Unit circle relation
Tangent & Secant	$\tan^{2} θ + 1 = \sec^{2} θ$	Divided by $\cos^{2} θ$
Double Angle (Sine)	$\sin 2 θ = 2 \sin θ \cos θ$	Sine duplication formula
Double Angle (Cosine)	$\cos 2 θ = \cos^{2} θ - \sin^{2} θ$	Cosine duplication formula

Angles and Radian Measure

In calculus, angles are measured in radians. If a central angle $θ$ in a circle of radius r intercepts an arc of length s, the radian measure of $θ$ is defined as:

θ = \frac{s}{r} ⟺ s = r θ

Central angle theta in circle of radius r intercepting arc s — Arc length $s = r θ$ and sector area $A = \frac{1}{2} r^{2} θ$ for central angle $θ$ in radians.

Since a full circle of radius r has circumference $2 π r$ , one full rotation of $360^{\circ}$ equals $2 π$ radians:

360^{\circ} = 2 π radians ⟺ 180^{\circ} = π radians

Degrees	$0^{\circ}$	$30^{\circ}$	$45^{\circ}$	$60^{\circ}$	$90^{\circ}$	$120^{\circ}$	$135^{\circ}$	$150^{\circ}$	$180^{\circ}$	$270^{\circ}$	$360^{\circ}$
Radians	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{3 π}{4}$	$\frac{5 π}{6}$	$π$	$\frac{3 π}{2}$	$2 π$

Directed Angles

An angle generated by rotating an initial side to a terminal side:

Positive angle: Counterclockwise rotation.
Negative angle: Clockwise rotation.

Positive counterclockwise angle vs negative clockwise angle — Positive angle (counterclockwise) and negative angle (clockwise).

Angles in Standard Position

An angle is in standard position when its vertex is at the origin $(0, 0)$ and its initial side lies along the positive x-axis.

Basic Trigonometric Functions

For an acute angle $θ$ ( $0 < θ < π / 2$ ) in a right triangle:

\sin θ = \frac{opposite}{hypotenuse}, \cos θ = \frac{adjacent}{hypotenuse}, \tan θ = \frac{opposite}{adjacent}

Right triangle with labeled sides relative to angle theta — Right triangle definition for acute angle $θ$ .

$θ$	$\sin θ$	$\cos θ$	$\tan θ$
$0$	$\frac{\sqrt{0}}{2} = 0$	$\frac{\sqrt{4}}{2} = 1$	$0$
$30^{\circ} = \frac{π}{6}$	$\frac{\sqrt{1}}{2} = \frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^{\circ} = \frac{π}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^{\circ} = \frac{π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{1}}{2} = \frac{1}{2}$	$\sqrt{3}$
$90^{\circ} = \frac{π}{2}$	$\frac{\sqrt{4}}{2} = 1$	$\frac{\sqrt{0}}{2} = 0$	Undefined

For arbitrary angles on a unit circle ( $r = 1$ ), if the terminal side intersects at $P (x, y)$ :

Unit circle trigonometric functions:

\sin θ = y, \cos θ = x, \tan θ = \frac{y}{x} (x \neq 0)

\sec θ = \frac{1}{x}, \csc θ = \frac{1}{y}, \cot θ = \frac{x}{y}

Unit circle definition of cos as x coordinate and sin as y coordinate — Unit circle definition: $\cos θ = x$ and $\sin θ = y$ .

Trigonometric Identities

Pythagorean Identities:

Even-Odd Identities:

\sin (- θ) = - \sin θ, \cos (- θ) = \cos θ, \tan (- θ) = - \tan θ

Addition Formulas:

\begin{aligned}
\sin(A+B) &= \sin A\cos B + \cos A\sin B \\
\cos(A+B) &= \cos A\cos B - \sin A\sin B \\
\tan(A+B) &= \frac{\tan A + \tan B}{1 - \tan A\tan B}
\end{aligned}

Double Angle Formulas:

\sin 2 θ = 2 \sin θ \cos θ

\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 1 - 2 \sin^{2} θ = 2 \cos^{2} θ - 1

Formulas for Lowering Powers:

\sin^{2} θ = \frac{1 - \cos 2 θ}{2}, \cos^{2} θ = \frac{1 + \cos 2 θ}{2}

Complementary Angle Identities:

Complementary angles theta and pi/2 minus theta in right triangle — Complementary angles $θ$ and $\frac{π}{2} - θ$ .

Laws of Cosines and Sines: For a triangle with interior angles A, B, C and sides a, b, c:

Triangle for Law of Sines and Law of Cosines — Triangle for the Law of Cosines and Law of Sines.