Trigonometry and its Applications
Trigonometry and its Applications covers the unit circle, radian measure, identities, trigonometric equations and graphs, inverse trigonometric functions, and triangle solving with the laws of sines and cosines. The exam tests fluency with exact values plus the discipline of keeping every solution inside a stated interval.
Core rules
- Radian measure: π radians equals 180 degrees. Arc length s = rθ and sector area A = (1/2)r^2 θ when θ is in radians.
- Unit circle: the point at angle θ is (cos θ, sin θ). Exact values at multiples of 30, 45, and 60 degrees come from reference angles plus the correct quadrant signs.
- Identities: sin^2 θ + cos^2 θ = 1, tan θ = sin θ/cos θ, 1 + tan^2 θ = sec^2 θ, sin 2θ = 2 sin θ cos θ, and cos 2θ = 1 - 2 sin^2 θ.
- Graphs: y = A sin(B(x - C)) + D has amplitude |A|, period 2π/|B|, phase shift C, and vertical shift D.
- Inverse functions: arcsin has range [-π/2, π/2], arccos has range [0, π], arctan has range (-π/2, π/2).
- Law of sines sin A/a = sin B/b = sin C/c, with the ambiguous SSA case checked separately; law of cosines a^2 = b^2 + c^2 - 2bc cos A.
- Solving sin θ = k yields two families of solutions per full period; restrict them to the required interval.
Worked example
Solve 2 sin^2 x - sin x - 1 = 0 on [0, 2π). Factor as a quadratic in sin x: (2 sin x + 1)(sin x - 1) = 0, so sin x = -1/2 or sin x = 1. The equation sin x = 1 gives x = π/2. For sin x = -1/2 the reference angle is π/6, and sine is negative in quadrants III and IV, giving x = 7π/6 and x = 11π/6. The solution set is {π/2, 7π/6, 11π/6}.
Traps that cost marks
- Leaving the calculator in degree mode when the answer must be in radians.
- Reporting only one solution family for sin θ = k and losing the other.
- Failing to rescale the interval when the argument is Bx rather than x.
- Applying the law of sines to an SSA triangle and reporting only one of the two possible triangles.
- Treating sin-inverse as the reciprocal 1/sin instead of the inverse function.