Study Notes
Trigonometry involves understanding the relationships and properties of trigonometric functions and using identities to simplify expressions and solve equations.
- Cosecant (cosec) — the reciprocal of sine. Example: If sin(θ) = 1/2, then cosec(θ) = 2.
- Secant (sec) — the reciprocal of cosine. Example: If cos(θ) = 1/3, then sec(θ) = 3.
- Cotangent (cot) — the reciprocal of tangent. Example: If tan(θ) = 4, then cot(θ) = 1/4.
- Compound Angle Formulae — used to find the sine, cosine, or tangent of the sum or difference of two angles. Example: sin(A ± B) = sinA cosB ± cosA sinB.
- Double Angle Formulae — used to express trigonometric functions of double angles. Example: sin(2A) = 2sinA cosA.
- Trigonometric Identities — equations involving trigonometric functions that are true for all values of the variables. Example: sin²(θ) + cos²(θ) = 1.
- Expressing a sinθ + b cosθ — converting expressions into the form R sin(θ ± α) or R cos(θ ± α). Example: a sinθ + b cosθ = R sin(θ + α), where R = √(a² + b²).
Exam Tips
Key Definitions to Remember
- Cosecant is the reciprocal of sine.
- Secant is the reciprocal of cosine.
- Cotangent is the reciprocal of tangent.
- Compound angle formulae for sine, cosine, and tangent.
- Double angle formulae for sine, cosine, and tangent.
Common Confusions
- Mixing up the reciprocal functions: cosec, sec, and cot.
- Incorrectly applying compound angle formulae.
- Forgetting to square root when finding R in expressions like a sinθ + b cosθ.
Typical Exam Questions
- What is the cosecant of an angle if sin(θ) = 0.5? Answer: Cosec(θ) = 2.
- How do you express a sinθ + b cosθ in the form R sin(θ + α)? Answer: R = √(a² + b²), α = arctan(b/a).
- Prove the identity sin²(θ) + cos²(θ) = 1. Answer: Use the Pythagorean identity.
What Examiners Usually Test
- Understanding and application of reciprocal trigonometric functions.
- Ability to use and derive compound and double angle formulae.
- Simplification of trigonometric expressions using identities.
- Conversion of expressions into the form R sin(θ ± α) or R cos(θ ± α).