Trigonometric Identities
Mastering identities is the key to simplifying complex expressions and solving higher-level equations.
๐ฏ By the end of this lesson you will be able to:
- Apply reciprocal and quotient identities to simplify expressions
- Use the three Pythagorean identities to rewrite trig expressions
- Convert expressions to sine and cosine to simplify complex problems
- Solve multi-step simplification problems using combined identities
๐ In This Lesson
1. Reciprocal & Quotient Identities
๐ข Identities to Memorize
- Reciprocal: \(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta = \frac{1}{\tan\theta}\)
- Quotient: \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)
โ๏ธ Practice Problems
1. Simplify: \(\sin\theta \cdot \csc\theta\)
๐๏ธ Show Solution
\(\sin\theta \cdot \frac{1}{\sin\theta} = 1\)
2. Simplify: \(\dfrac{\cot\theta}{\cos\theta}\)
๐๏ธ Show Solution
\(\dfrac{\cos\theta / \sin\theta}{\cos\theta} = \dfrac{1}{\sin\theta} = \csc\theta\)
3. Simplify: \(\tan\theta \cdot \cos\theta\)
๐๏ธ Show Solution
\(\dfrac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta\)
๐ฌ Watch: Simplifying with Reciprocal & Quotient Identities
2. Pythagorean Identities
๐ข The Three Pythagorean Identities
1. \(\sin^2\theta + \cos^2\theta = 1\)
2. \(1 + \tan^2\theta = \sec^2\theta\)
3. \(1 + \cot^2\theta = \csc^2\theta\)
โ๏ธ Practice Problems
4. Simplify: \(1 – \cos^2\theta\)
๐๏ธ Show Solution
Since \(\sin^2\theta + \cos^2\theta = 1\), it follows that \(1 – \cos^2\theta = \sin^2\theta\)
5. Simplify: \(\sec^2\theta – 1\)
๐๏ธ Show Solution
From \(1 + \tan^2\theta = \sec^2\theta\), subtract 1: \(\sec^2\theta – 1 = \tan^2\theta\)
6. Simplify: \((\csc^2\theta – 1)\tan^2\theta\)
๐๏ธ Show Solution
\((\cot^2\theta)\tan^2\theta = \left(\dfrac{1}{\tan^2\theta}\right)\tan^2\theta = 1\)
3. Advanced Simplifying Strategy
๐ก The “Sine-Cosine” Rule
When you get stuck, convert all functions (tangent, secant, etc.) into terms of Sine and Cosine. This often makes the path to the answer much clearer!
โ๏ธ Practice Problems
7. Simplify: \(\dfrac{\sin^2\theta}{1 – \cos\theta}\)
๐๏ธ Show Solution
Use \(1 – \cos^2\theta = \sin^2\theta\). Then:
\(\dfrac{1 – \cos^2\theta}{1 – \cos\theta} = \dfrac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta} = 1 + \cos\theta\)
8. Simplify: \(\tan\theta \cdot \csc\theta\)
๐๏ธ Show Solution
\(\dfrac{\sin\theta}{\cos\theta} \cdot \dfrac{1}{\sin\theta} = \dfrac{1}{\cos\theta} = \sec\theta\)
9. Simplify: \(\dfrac{\sec\theta}{\tan\theta}\)
๐๏ธ Show Solution
\(\dfrac{1/\cos\theta}{\sin\theta/\cos\theta} = \dfrac{1}{\sin\theta} = \csc\theta\)
๐ฌ Watch: Advanced Simplifying โ converting to Sine & Cosine
