1. What Are Complementary Angles?
Two angles are called complementary if their sum is 90°.
For example: 30° and 60°, 45° and 45°, 20° and 70°.
In trigonometry, the most common complementary angle expressions are:
\( 90^\circ - \theta \)
We compare the values of ratios at angle \(\theta\) with the values at \(90^\circ - \theta\).
2. Complementary Angle Identities
Trigonometric ratios of complementary angles swap between sin and cos, tan and cot, sec and cosec.
Here are the key identities:
\( \sin(90^\circ - \theta) = \cos \theta \)
\( \cos(90^\circ - \theta) = \sin \theta \)
\( \tan(90^\circ - \theta) = \cot \theta \)
\( \cot(90^\circ - \theta) = \tan \theta \)
\( \sec(90^\circ - \theta) = \csc \theta \)
\( \csc(90^\circ - \theta) = \sec \theta \)
2.1. Reason Behind the Swap
In a right triangle, for an acute angle \(\theta\):
- Opposite side to \(\theta\) becomes the adjacent side to \(90^\circ-\theta\)
- Adjacent side to \(\theta\) becomes the opposite side to \(90^\circ-\theta\)
So sin and cos interchange, tan and cot interchange, and sec and cosec interchange.
3. Table of Complementary Angle Values
This table shows how values match when angles add up to 90°.
| Expression | Equivalent Ratio |
|---|---|
| \( \sin(90^\circ - \theta) \) | \( \cos \theta \) |
| \( \cos(90^\circ - \theta) \) | \( \sin \theta \) |
| \( \tan(90^\circ - \theta) \) | \( \cot \theta \) |
| \( \cot(90^\circ - \theta) \) | \( \tan \theta \) |
| \( \sec(90^\circ - \theta) \) | \( \csc \theta \) |
| \( \csc(90^\circ - \theta) \) | \( \sec \theta \) |
4. Examples to Understand Complementary Ratios
Example 1: Find \( \sin 30^\circ \) using complementary identity.
- \(30^\circ = 90^\circ - 60^\circ\)
\( \sin 30^\circ = \cos 60^\circ = 1/2 \)
Example 2: Evaluate \( \tan 20^\circ \) using \( \cot \).
- \(20^\circ = 90^\circ - 70^\circ\)
\( \tan 20^\circ = \cot 70^\circ \)
Example 3: If \( \sec 35^\circ = k \), then:
\( \csc 55^\circ = k \)
because \(55^\circ = 90^\circ - 35^\circ\).