Solving Trigonometric Equations

This lesson brings together all the techniques for solving trigonometric equations at A-Level (9MA0): finding solutions in given ranges, handling multiple solutions, using identities to simplify, and understanding general solutions.

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Strategy for Solving Trig Equations

1. Simplify using identities if needed (to get a single trig function).
2. Rearrange to isolate the trig function.
3. Find the principal value using inverse trig or exact values.
4. Find all solutions in the given range using CAST diagram or graph.

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The CAST Diagram

The CAST diagram tells you in which quadrants each trig function is positive:

Quadrant	Angle range	Positive functions
1st	0 to pi/2	All (sin, cos, tan)
2nd	pi/2 to pi	Sin only
3rd	pi to 3pi/2	Tan only
4th	3pi/2 to 2pi	Cos only

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Solving Basic Equations

Worked Example 1

Solve sin(x) = sqrt(3)/2 for 0 <= x <= 2pi.

Principal value: x = pi/3

Using CAST: sin is positive in Q1 and Q2.

Q1: x = pi/3 Q2: x = pi - pi/3 = 2pi/3