1. What Are Trigonometric Identities?
A trigonometric identity is an equation involving trigonometric ratios that is true for all valid values of the angle \(\theta\). These identities form the foundation of most trigonometry problems, simplifications and proofs.
The three most important identities come directly from the Pythagorean Theorem.
2. Identity 1: sin²θ + cos²θ = 1
This is the most basic and fundamental identity in trigonometry. It originates from the Pythagorean relation in a right triangle:
\( (\text{Opp})^2 + (\text{Adj})^2 = (\text{Hyp})^2 \)
Dividing all terms by \((\text{Hyp})^2\), we get:
\( \sin^2 \theta + \cos^2 \theta = 1 \)
2.1. Example
If \( \sin \theta = \dfrac{3}{5} \), find \( \cos \theta \).
\( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \dfrac{9}{25} = \dfrac{16}{25} \)
\( \cos \theta = \dfrac{4}{5} \)
3. Identity 2: 1 + tan²θ = sec²θ
This identity is obtained by dividing the main Pythagorean identity by \(\cos^2 \theta\):
\( \dfrac{\sin^2 \theta}{\cos^2 \theta} + 1 = \dfrac{1}{\cos^2 \theta} \)
\( \tan^2 \theta + 1 = \sec^2 \theta \)
3.1. Example
If \( \tan \theta = 2 \), then:
\( \sec^2 \theta = 1 + \tan^2 \theta = 1 + 4 = 5 \)
\( \sec \theta = \sqrt{5} \)
4. Identity 3: 1 + cot²θ = cosec²θ
This identity comes from dividing the fundamental identity by \(\sin^2 \theta\):
\( \dfrac{\cos^2 \theta}{\sin^2 \theta} + 1 = \dfrac{1}{\sin^2 \theta} \)
\( \cot^2 \theta + 1 = \csc^2 \theta \)
4.1. Example
If \( \cot \theta = 3 \), then:
\( \csc^2 \theta = 1 + \cot^2 \theta = 1 + 9 = 10 \)
\( \csc \theta = \sqrt{10} \)
5. Using Identities to Rewrite Ratios
These identities allow us to convert between sin, cos, tan and their reciprocals easily. They are especially useful when simplifying trigonometric expressions or solving equations.
5.1. Example
If \( \sec \theta = 5 \), find \( \tan \theta \).
\( \sec^2 \theta = 25 \)
Using \( \sec^2 \theta = 1 + \tan^2 \theta \):
\( 25 = 1 + \tan^2 \theta \)
\( \tan^2 \theta = 24 \)
\( \tan \theta = \sqrt{24} = 2\sqrt{6} \)