Study Notes
Trigonometric identities and equations involve understanding the properties and relationships of trigonometric functions across all quadrants and using these identities to solve equations. This includes using exact values and simplifying expressions with identities.
- Angle in Quadrants — An angle is measured from the positive x-axis, with anticlockwise as positive and clockwise as negative. Example: An angle of -150° is in the 3rd quadrant with a basic angle of 30°.
- Trigonometric Ratios — Ratios like sine, cosine, and tangent have specific values for standard angles like 30°, 45°, and 60°. Example: sin 30° = 1/2, cos 45° = √2/2.
- Trigonometric Identities — Equations true for all variable values, often used to simplify expressions. Example: sin²θ + cos²θ ≡ 1.
- Solving Trigonometric Equations — Involves finding angle solutions within a given range using identities and graphs. Example: Solving for x in -360° < x < 360° with solutions like -45°, 105°.
Exam Tips
Key Definitions to Remember
- An angle in the 3rd quadrant is negative and between -180° and 0°.
- Trigonometric identities are equations true for all values of the variable.
Common Confusions
- Mixing up the signs of trigonometric functions in different quadrants.
- Forgetting to convert negative angles to their positive equivalents.
Typical Exam Questions
- What is the basic angle for -150°? 30°
- Solve sin x = 0.5 for -360° < x < 360°? x = 30°, 150°, -210°, -330°
- Prove the identity sin²θ + cos²θ = 1? Use the Pythagorean identity.
What Examiners Usually Test
- Ability to use trigonometric identities to simplify expressions.
- Solving equations for angles in all four quadrants.
- Understanding and applying exact values of trigonometric ratios.