Trigonometric Functions and Identities
Core concepts
You need the unit circle definitions of sine, cosine, and tangent, exact values at standard angles, radian measure, amplitude, period, phase shift, and the graphs of the reciprocal functions. For y = a sin(b(x − c)) + d, the amplitude is |a|, the period is 2π/|b|, the phase shift is c, and d is the vertical translation.
Identities are central: the Pythagorean identity sin²θ + cos²θ = 1, the double-angle formulas, and the compound-angle formulas. These let you simplify expressions and solve equations. Solving trigonometric equations requires finding all solutions in the given interval, usually by identifying a reference angle and then using symmetry in the four quadrants.
Traps to watch
- Losing solutions by dividing by a trigonometric factor instead of factorising.
- Forgetting that tan θ is undefined at odd multiples of π/2.
- Using degrees when the question specifies radians, or vice versa.
- Assuming sin(a + b) = sin a + sin b, which is false.
Rule of thumb
Factorise rather than divide when a trigonometric equation has several terms. Find the reference angle, then list all solutions in the stated interval before selecting.
Worked example
Solve 2 sin²θ − sin θ − 1 = 0 for 0 ≤ θ < 2π. Factor: (2 sin θ + 1)(sin θ − 1) = 0. So sin θ = −½ or sin θ = 1. The first gives θ = 7π/6 and 11π/6; the second gives θ = π/2. All four check in the original interval.