Complementary numbers, specifically in the context of angles, are pairs of angles that add up to 90 degrees.
Understanding Complementary Angles
The concept of complementary angles is crucial in trigonometry. Here's how to find them:
- Definition: Two angles are considered complementary if their sum equals 90 degrees.
- Formula: If one angle is represented by 'x', its complementary angle is '90° - x'.
Example
If an angle measures 30 degrees, its complementary angle is:
90° - 30° = 60°
Therefore, 30° and 60° are complementary angles.
Complementary Angles in Trigonometry
Complementary angles play a significant role in trigonometric ratios, where sine and cosine functions are related through the following identities:
| Trigonometric Identity | Explanation |
|---|---|
| sin (90° - A) = cos A | The sine of an angle's complement equals the cosine of the original angle. |
| cos (90° - A) = sin A | The cosine of an angle's complement equals the sine of the original angle. |
Practical Applications
- Solving Triangles: These relationships help solve right-angled triangles.
- Trigonometric Functions: It helps in understanding the interrelation between different trigonometric functions.
Finding the Complementary Number
To find the complementary number (or angle), subtract the given number from 90.
- Given Angle: If you have one angle's measure, you can easily determine its complement by subtracting it from 90 degrees.
Example:
If you have an angle of 40 degrees, the complementary angle will be 90 - 40 = 50 degrees.
Summary
- Complementary angles add to 90 degrees.
- The complement of 'x' is '90° - x'.
- Trigonometric relationships: sin(90° - A) = cos A and cos(90° - A) = sin A
