Trigonometry — II
Easy Overview
So you know what sin and cos are. Cool. But what if I told you sin(A + B) isn't sin A + sin B? (Yeah, that's a common mistake.) This chapter takes trig to the next level — adding angles, doubling them, splitting them, and even reversing the functions to find angles from ratios.
Compound angle formulas
Here's where the magic happens. sin(A + B) = sin A cos B + cos A sin B. And sin(A − B) = sin A cos B − cos A sin B. Cos is similar but with a twist: cos(A + B) = cos A cos B − sin A sin B. Don't just memorise them — notice the pattern. Sin is symmetrical (sin-cos then cos-sin), cos switches between cos and sin with a minus in the middle. Practise a few and they'll stick.
Double angle formulas
Just set B = A in the compound formulas and you get: sin 2A = 2 sin A cos A. And cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A. That last form (cos 2A = 2 cos²A − 1) is super useful for integration later. These also let you derive half-angle and triple-angle formulas if you need 'em.
Trigonometric equations
Solving sin θ = 0.5 means finding ALL angles where that's true, not just one. General formula: If sin θ = k, then θ = nπ + (−1)ⁿ·α where α is the principal solution. For cos θ = k, it's θ = 2nπ ± α. For tan θ = k, it's θ = nπ + α. The n ∈ Z means 'any integer', so you capture every possible answer across infinite rotations.
Inverse trigonometric functions
Sometimes you know the ratio but need the angle. That's sin⁻¹, cos⁻¹, tan⁻¹. They're called inverse functions. But careful — sin⁻¹(0.5) doesn't give you infinite answers; it gives the 'principal value', which is the simplest one. For sin⁻¹, the output is between −90° and 90°. For cos⁻¹, it's between 0° and 180°. Domain restrictions make them proper functions.
Key Points
- •sin(A ± B) = sin A cos B ± cos A sin B
- •cos(A ± B) = cos A cos B ∓ sin A sin B
- •sin 2A = 2 sin A cos A
- •cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
- •General solutions: sin θ = k → θ = nπ + (−1)ⁿα
- •Inverse functions return principal values only
- •Domain of sin⁻¹ is [−1, 1], range is [−π/2, π/2]
Practice Questions
- Prove that sin(A + B) sin(A − B) = sin²A − sin²B.
- If tan A = 1/2 and tan B = 1/3, find tan(A + B). What can you say about A + B?
- Solve sin θ + cos θ = 1 for 0° ≤ θ ≤ 360°.
- Prove that (sin 3θ)/(sin θ) − (cos 3θ)/(cos θ) = 2.
- Find the principal value of sin⁻¹(−1/2) and tan⁻¹(√3).