# 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.

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## 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

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## 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.

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## 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:**  \( \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \)

\( \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \)

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## 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°) | - | + | - |

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## 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.

\[ h = 30 \times \tan 60° = 30\sqrt{3} \approx 51.96 \text{ m} \]

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## 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.