Trigonometry
Trigonometry tests radian measure, the reciprocal and Pythagorean identities, solving trigonometric equations over a stated interval, and combining sine and cosine into a single harmonic term. You should be fluent in both degrees and radians and know the exact values for the standard angles.
Radians, arcs and sectors
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. For an angle θ in radians: arc length = rθ, sector area = ½r²θ, and the area of a triangle formed by two radii is ½r² sin θ. Always set the calculator to the mode matching the question.
Identities and equations
Core identities: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ. Dividing the Pythagorean identity by cos²θ or sin²θ produces the sec–tan and cosec–cot forms. For equations, use the symmetries of the graphs or a CAST diagram to find all solutions in the required interval; the sine and cosine curves repeat every 2π and tangent every π.
Harmonic form
An expression a sin θ + b cos θ can be written as R sin(θ + α) or R cos(θ − α), where R = √(a² + b²) and α satisfies cos α = a/R and sin α = b/R. This is used to find maximum and minimum values and to solve equations.
Worked example
Solve 3 sin θ + 4 cos θ = 2 for 0 ≤ θ < 2π.
Here R = √(9 + 16) = 5, and α = arctan(4/3) ≈ 0.927 rad, so 5 sin(θ + 0.927) = 2, giving sin(θ + 0.927) = 0.4. The solutions of sin X = 0.4 are X ≈ 0.412 and X ≈ π − 0.412 ≈ 2.730. Hence θ ≈ −0.515 (invalid) or θ ≈ 1.803, and adding 2π to the first gives θ ≈ 5.768. The solutions in the interval are θ ≈ 1.80 and θ ≈ 5.77.
Traps that cost marks
- Losing solutions by dividing by a trigonometric factor instead of factorising.
- Working in degrees when the interval is given in radians.
- Squaring an equation and failing to reject extraneous roots.
- Choosing the wrong quadrant when taking an inverse sine or cosine.