Welcome to the magical world of Trigonometry! This is the branch of mathematics that studies relationships between side lengths and angles of triangles. The word comes from the Greek words "trigonon" (triangle) and "metron" (measure).
In this journey, you'll learn how to:
Some angles have special trigonometric values that are worth memorizing. Let's explore them interactively!
These angles have exact values that can be derived from geometric properties:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | ∞ |
Two angles are complementary if they add up to 90°. There's a special relationship between their trigonometric ratios!
For any acute angle θ, the following relationships hold:
This happens because in a right-angled triangle, the two non-right angles are complementary. The opposite side for one angle becomes the adjacent side for the other angle.
Before calculators, trigonometric tables were essential tools. Let's learn how to use them effectively!
Trigonometric tables list angles (usually from 0° to 90°) in increments (often 1° or 6') with their corresponding sine, cosine, and tangent values.
Locate the angle in the leftmost column, then read across to find the desired ratio.
Let's look at a sample trigonometric table:
| Angle | sin | cos |
|---|---|---|
| 0° | 0.0000 | 1.0000 |
| 30° | 0.5000 | 0.8660 |
| 45° | 0.7071 | 0.7071 |
| 60° | 0.8660 | 0.5000 |
| 90° | 1.0000 | 0.0000 |
For angles between those listed, we can estimate values using interpolation.
Try finding these values: