Trigonometry Basics — Sin, Cos, Tan and Essential Identities
Sin, cos, and tan aren't just three hard-to-remember letters — they're tools for solving a wide range of problems involving triangles, angles, and real-world applications. Learn them once, use them forever.
MathPal Team
February 28, 2026
3 min read
Trigonometry is a topic many students "learn and forget," mostly because they memorize tables of values without understanding the underlying concepts. This post helps you understand it correctly — and remember it for good.
What is trigonometry?
Trigonometry studies the relationship between the sides and angles of triangles. It starts with right triangles, then extends to the unit circle and all real numbers.
Definitions in a right triangle
For a right triangle with acute angle ( \alpha ), opposite side ( a ), adjacent side ( b ), and hypotenuse ( c ):
[ \sin\alpha = \frac{a}{c}, \quad \cos\alpha = \frac{b}{c}, \quad \tan\alpha = \frac{a}{b} ]
[ \cot\alpha = \frac{b}{a} = \frac{1}{\tan\alpha} ]
Memory trick — SOH-CAH-TOA:
- S in = O pposite / H ypotenuse
- C os = A djacent / H ypotenuse
- T an = O pposite / A djacent
Values at special angles
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | ( \frac{1}{2} ) | ( \frac{\sqrt{2}}{2} ) | ( \frac{\sqrt{3}}{2} ) | 1 |
| cos | 1 | ( \frac{\sqrt{3}}{2} ) | ( \frac{\sqrt{2}}{2} ) | ( \frac{1}{2} ) | 0 |
| tan | 0 | ( \frac{\sqrt{3}}{3} ) | 1 | ( \sqrt{3} ) | undefined |
Memory trick for sin row: 0, 1/2, ( \sqrt{2}/2 ), ( \sqrt{3}/2 ), 1 — the numerators are 0,1,2,3,4 divided by 2. The cos row is just the reverse.
Fundamental trigonometric identities
Pythagorean identity: ( \sin^2\alpha + \cos^2\alpha = 1 )
Derived from it: ( 1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}, \quad 1 + \cot^2\alpha = \frac{1}{\sin^2\alpha} )
Sum and difference formulas: ( \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta )
( \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta )
Trigonometry on the unit circle
When angle ( \alpha ) is no longer limited to [0°, 90°], we use the unit circle (radius = 1) to extend the definitions:
- A point on the circle has coordinates ( (\cos\alpha, \sin\alpha) )
- Angles increase counterclockwise
- The signs of sin/cos depend on the quadrant
| Quadrant | sin | cos | tan |
|---|---|---|---|
| I (0°–90°) | + | + | + |
| II (90°–180°) | + | - | - |
| III (180°–270°) | - | - | + |
| IV (270°–360°) | - | + | - |
Real-world applications
- Indirect measurement: Know the distance and angle of elevation, calculate the height of a building
- Physics: Decompose forces into horizontal and vertical components
- Computer graphics: Rotate objects in 2D/3D space
Example: A person stands 30 m from the base of a building. The angle of elevation to the top is ( 60° ). Find the building's height.
[ h = 30 \times \tan 60° = 30\sqrt{3} \approx 51.96 \text{ m} ]
Practice with MathPal
Take a photo of your trigonometry problem and upload it to MathPal — AI will identify the angle, apply the right formula, and walk you through every step with clear diagrams.