Trigonometry Essentials: Sin, Cos, Tan and Key Identities

2026.04.14

Learn.byAuthor

Trigonometry connects angles to side lengths in triangles — and extends far beyond into waves, circles, and calculus. It’s essential for every math student from high school through university.

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Key Point: Trigonometry shows up on the SAT (2~3 questions), ACT (4~6 questions), and AP Calculus. The unit circle and basic identities are the foundation for everything else.

The Three Basic Ratios

Definition

For a right triangle with angle $\theta$ :

\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}

Memory trick: SOH-CAH-TOA

Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

Key Angle Values

Angle	sin	cos	tan
$0\degree$	$0$	$1$	$0$
$30\degree$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45\degree$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60\degree$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90\degree$	$1$	$0$	undefined

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. For any angle $\theta$ :

(\cos\theta,\; \sin\theta) = \text{coordinates on the unit circle}

This extends trig beyond right triangles to any angle, including negative angles and angles greater than $360\degree$ .

Essential Identities

Pythagorean Identity

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

Divide by $\cos^2\theta$ : $\tan^2\theta + 1 = \sec^2\theta$

Divide by $\sin^2\theta$ : $1 + \cot^2\theta = \csc^2\theta$

Double Angle Formulas

\sin 2\theta = 2\sin\theta\cos\theta

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

Addition Formulas

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

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

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Watch Out: In the cosine addition formula, the sign flips — plus becomes minus and vice versa. This is the opposite of the sine formula and a common exam trap.

Graphs of Trig Functions

$y = \sin x$ : Period $2\pi$ , amplitude 1, starts at origin
$y = \cos x$ : Period $2\pi$ , amplitude 1, starts at maximum
$y = \tan x$ : Period $\pi$ , vertical asymptotes at $x = \frac{\pi}{2} + n\pi$

For $y = A\sin(Bx + C) + D$ :

$|A|$ = amplitude
$\frac{2\pi}{|B|}$ = period
$-\frac{C}{B}$ = phase shift
$D$ = vertical shift

Worked Examples

Example 1: Find an Exact Value

Find $\sin 75\degree$ using the addition formula.

Show Solution

$\sin 75\degree = \sin(45\degree + 30\degree)$

= \sin 45\degree\cos 30\degree + \cos 45\degree\sin 30\degree

= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

Example 2: Solve a Trig Equation

Solve $2\sin\theta - 1 = 0$ for $0 \leq \theta < 2\pi$ .

Show Solution

$\sin\theta = \frac{1}{2}$

$\theta = \frac{\pi}{6}$ or $\theta = \frac{5\pi}{6}$

Example 3: Pythagorean Identity

If $\sin\theta = \frac{3}{5}$ and $\theta$ is in the first quadrant, find $\cos\theta$ .

Show Solution

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

$\frac{9}{25} + \cos^2\theta = 1$

$\cos^2\theta = \frac{16}{25}$ → $\cos\theta = \frac{4}{5}$ (positive in Q1)

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Pro Tip: Memorize the unit circle values for

0\degree, 30\degree, 45\degree, 60\degree, 90\degree

. Every other angle on the unit circle is just a reflection of these five values.

Top 3 Common Mistakes

Mixing up sine and cosine on the unit circle — x-coordinate is cosine, y-coordinate is sine
Forgetting the sign flip in cosine addition formula — $\cos(A + B)$ uses minus, not plus
Not checking all solutions in a trig equation — trig functions are periodic, so there are usually multiple solutions in $[0, 2\pi)$

Trigonometry Essentials: Sin, Cos, Tan and Key Identities

The Three Basic Ratios

Definition

Key Angle Values

The Unit Circle

Essential Identities

Pythagorean Identity

Double Angle Formulas

Addition Formulas

Graphs of Trig Functions

Worked Examples

Example 1: Find an Exact Value

Example 2: Solve a Trig Equation

Example 3: Pythagorean Identity

Top 3 Common Mistakes

Related Topics