Trigonometric Functions
Easy Overview
Remember sine, cosine, and tan from earlier? This chapter takes them to the next level. You'll learn identities that connect them all, solve trigonometric equations, and work with angles that go beyond 90°. It's the toolkit you need for physics, engineering, and making sense of waves.
Compound Angle Formulas
Ever wondered what sin(A + B) is? It's NOT sin A + sin B — that's the biggest mistake students make. The real formulas are: sin(A + B) = sin A cos B + cos A sin B. Think of it like a dance move — sine and cosine switching partners. You just need to memorize these; there's no shortcut.
Double and Triple Angle Formulas
These come straight from the compound formulas. If B = A, then sin 2A = 2 sin A cos A. Simple, right? And cos 2A has THREE forms: cos²A − sin²A, 2 cos²A − 1, and 1 − 2 sin²A. Pick whichever one helps you solve the problem. Triple angles are the same idea but messier — sin 3A = 3 sin A − 4 sin³A.
Factorization Formulas
These let you turn sums into products and vice versa. sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2). Why bother? Because sometimes the product form is easier to solve. It's like simplifying a fraction — you change the shape to see the answer better.
Trigonometric Equations
Solving sin θ = 1/2 is not just 'θ = 30°'. No — there are INFINITE solutions because trig functions repeat. The general solution is θ = nπ + (−1)ⁿ(π/6). For sin and cos, solutions repeat every 2π. For tan, every π. Always add the period to your answer.
Inverse Trigonometric Functions
If sin θ = x, then θ = sin⁻¹ x. It's like asking 'what angle gives me this value?' But there's a catch — multiple angles give the same value, so we restrict the range. sin⁻¹ gives values between −π/2 and π/2. cos⁻¹ gives between 0 and π. Just remember which quadrant you're stuck in.
Key Points
- •sin(A + B) = sin A cos B + cos A sin B — memorize this
- •cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
- •sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2)
- •General solution for sin θ = sin α: θ = nπ + (−1)ⁿα
- •General solution for cos θ = cos α: θ = 2nπ ± α
- •General solution for tan θ = tan α: θ = nπ + α
- •Principal values: sin⁻¹ ∈ [−π/2, π/2], cos⁻¹ ∈ [0, π]
Practice Questions
- Prove that sin 3A = 3 sin A − 4 sin³A.
- Solve the equation 2 cos²θ − √3 cos θ = 0 for θ ∈ [0, 2π].
- Find the general solution of tan θ + cot θ = 2.
- If sin A = 3/5 and cos B = 5/13, find sin(A + B) where A and B are acute.