Key Idea: This topic adds sec, csc and cot, and the inverse functions arcsin, arccos and arctan. You use them to solve equations with exact answers on Paper 1, or with the GDC on Paper 2.
Reciprocal trig functions
- sec = 1/cos, csc = 1/sin, cot = 1/tan.
- 1 + tan² = sec² (divide sin² + cos² = 1 by cos²).
- 1 + cot² = csc² (divide by sin²).
- To solve, rewrite in sin, cos or tan first.
| Function | Inputs | Output angle |
|---|---|---|
| arcsin x | [−1, 1] | [−π/2, π/2] |
| arccos x | [−1, 1] | [0, π] |
| arctan x | all x | (−π/2, π/2) |
Composing and solving
- For cos(arcsin x), set θ = arcsin x and build a right triangle.
- For arccos(2x) = π/3, apply cos to both sides, then check the input lies in [−1, 1].
Each inverse gives one angle from a fixed range.
Solve 2 sec x = 4 for 0 ≤ x < 2π. cos x = 1/2, so x = π/3 or x = 5π/3.
Which ratio is csc the reciprocal of? sin.
Solve arccos(2x) = π/3. 2x = 1/2, so x = 1/4.
Can arcsin give 2π/3? No. Its output stays between −π/2 and π/2.