Trigonometric Identities
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Complementary Angle Identities (Cofunction Identities)
Definition:
Complementary angles are two angles that sum to 90° or \( \frac{\pi}{2} \) radians
Cofunctions of complementary angles are equal
Cofunction Identities:
| In Radians | In Degrees |
|---|---|
| \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta \) | \( \sin(90° - \theta) = \cos\theta \) |
| \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta \) | \( \cos(90° - \theta) = \sin\theta \) |
| \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta \) | \( \tan(90° - \theta) = \cot\theta \) |
| \( \cot\left(\frac{\pi}{2} - \theta\right) = \tan\theta \) | \( \cot(90° - \theta) = \tan\theta \) |
| \( \sec\left(\frac{\pi}{2} - \theta\right) = \csc\theta \) | \( \sec(90° - \theta) = \csc\theta \) |
| \( \csc\left(\frac{\pi}{2} - \theta\right) = \sec\theta \) | \( \csc(90° - \theta) = \sec\theta \) |
Example:
Simplify: \( \sin(60°) \) using cofunction
\( \sin(60°) = \cos(90° - 60°) = \cos(30°) \)
2. Symmetry and Periodicity
Even and Odd Functions:
Even Functions (Symmetric about y-axis):
\[ \cos(-\theta) = \cos\theta \] \[ \sec(-\theta) = \sec\theta \]
Odd Functions (Symmetric about origin):
\[ \sin(-\theta) = -\sin\theta \] \[ \tan(-\theta) = -\tan\theta \] \[ \csc(-\theta) = -\csc\theta \] \[ \cot(-\theta) = -\cot\theta \]
Periodicity:
Period = smallest positive value for which \( f(\theta + p) = f(\theta) \)
Period \( 2\pi \):
• \( \sin(\theta + 2\pi) = \sin\theta \)
• \( \cos(\theta + 2\pi) = \cos\theta \)
• \( \csc(\theta + 2\pi) = \csc\theta \), \( \sec(\theta + 2\pi) = \sec\theta \)
Period \( \pi \):
• \( \tan(\theta + \pi) = \tan\theta \)
• \( \cot(\theta + \pi) = \cot\theta \)
3. Reciprocal Identities
\[ \sin\theta = \frac{1}{\csc\theta} \quad \text{and} \quad \csc\theta = \frac{1}{\sin\theta} \] \[ \cos\theta = \frac{1}{\sec\theta} \quad \text{and} \quad \sec\theta = \frac{1}{\cos\theta} \] \[ \tan\theta = \frac{1}{\cot\theta} \quad \text{and} \quad \cot\theta = \frac{1}{\tan\theta} \]
4. Quotient Identities
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \] \[ \cot\theta = \frac{\cos\theta}{\sin\theta} \]
5. Pythagorean Identities
The Three Pythagorean Identities:
\[ \sin^2\theta + \cos^2\theta = 1 \]
This is the fundamental Pythagorean identity
\[ 1 + \tan^2\theta = \sec^2\theta \]
Divide first identity by \( \cos^2\theta \)
\[ 1 + \cot^2\theta = \csc^2\theta \]
Divide first identity by \( \sin^2\theta \)
Alternate Forms:
From \( \sin^2\theta + \cos^2\theta = 1 \):
• \( \sin^2\theta = 1 - \cos^2\theta \)
• \( \cos^2\theta = 1 - \sin^2\theta \)
From \( 1 + \tan^2\theta = \sec^2\theta \):
• \( \tan^2\theta = \sec^2\theta - 1 \)
• \( \sec^2\theta - \tan^2\theta = 1 \)
From \( 1 + \cot^2\theta = \csc^2\theta \):
• \( \cot^2\theta = \csc^2\theta - 1 \)
• \( \csc^2\theta - \cot^2\theta = 1 \)
Example:
Given \( \sin\theta = \frac{3}{5} \), find \( \cos\theta \)
Use \( \sin^2\theta + \cos^2\theta = 1 \)
\( \left(\frac{3}{5}\right)^2 + \cos^2\theta = 1 \)
\( \frac{9}{25} + \cos^2\theta = 1 \)
\( \cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25} \)
\( \cos\theta = \pm\frac{4}{5} \)
6. Sum and Difference Identities
Sine Identities:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
Cosine Identities:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Tangent Identities:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Example 1:
Find exact value: \( \sin(75°) \)
Write as: \( \sin(75°) = \sin(45° + 30°) \)
Use sum formula: \( \sin(45°)\cos(30°) + \cos(45°)\sin(30°) \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)
\( = \frac{\sqrt{6} + \sqrt{2}}{4} \)
Example 2:
Find exact value: \( \cos(15°) \)
Write as: \( \cos(15°) = \cos(45° - 30°) \)
Use difference formula: \( \cos(45°)\cos(30°) + \sin(45°)\sin(30°) \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)
\( = \frac{\sqrt{6} + \sqrt{2}}{4} \)
7. Using Multiple Identities
Strategy:
1. Start with given information
2. Use Pythagorean identities to find related ratios
3. Apply reciprocal or quotient identities as needed
4. Consider quadrant to determine signs
Example:
Given \( \tan\theta = 2 \) and \( \theta \) is in Quadrant III, find all six trig ratios
Step 1: Use \( 1 + \tan^2\theta = \sec^2\theta \)
\( 1 + 4 = \sec^2\theta \) → \( \sec\theta = -\sqrt{5} \) (negative in Q3)
Step 2: \( \cos\theta = \frac{1}{\sec\theta} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5} \)
Step 3: \( \sin\theta = \tan\theta \cdot \cos\theta = 2 \cdot (-\frac{\sqrt{5}}{5}) = -\frac{2\sqrt{5}}{5} \)
Step 4: Find remaining ratios using reciprocals
\( \csc\theta = -\frac{\sqrt{5}}{2} \), \( \cot\theta = \frac{1}{2} \)
8. Solve Trigonometric Equations Using Identities
Steps:
1. Use identities to simplify the equation
2. Get all terms on one side (= 0)
3. Factor if possible
4. Solve for the trig function
5. Find all solutions in the given interval
Example:
Solve: \( 2\sin^2\theta = \cos\theta + 1 \) for \( 0 \leq \theta < 2\pi \)
Use \( \sin^2\theta = 1 - \cos^2\theta \):
\( 2(1 - \cos^2\theta) = \cos\theta + 1 \)
\( 2 - 2\cos^2\theta = \cos\theta + 1 \)
\( -2\cos^2\theta - \cos\theta + 1 = 0 \)
\( 2\cos^2\theta + \cos\theta - 1 = 0 \)
Factor: \( (2\cos\theta - 1)(\cos\theta + 1) = 0 \)
\( \cos\theta = \frac{1}{2} \) or \( \cos\theta = -1 \)
Solutions: \( \theta = \frac{\pi}{3}, \pi, \frac{5\pi}{3} \)
9. Quick Reference Summary
Essential Identities:
Pythagorean: \( \sin^2\theta + \cos^2\theta = 1 \)
Cofunction: \( \sin(90° - \theta) = \cos\theta \)
Sum: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
Difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
Even: \( \cos(-\theta) = \cos\theta \)
Odd: \( \sin(-\theta) = -\sin\theta \)
📚 Study Tips
✓ Memorize fundamental Pythagorean identity - derive others from it
✓ Cofunctions: sine ↔ cosine, tan ↔ cot, sec ↔ csc
✓ Sum/difference formulas: Note sign changes between formulas
✓ Always consider the quadrant when finding trig ratios
✓ Check your solutions in the original equation