IGCSE Additional Mathematics (0606)

Solving trigonometric equations

What this topic tests, the mistakes that cost marks, and one exam-style question worked through step by step.

Key ideas and common mistakes

When solving a trigonometric equation set as a product equal to zero, solve each factor separately and check that all solutions fall within the specified interval.
Common mistake: Students often miss solutions by not considering both factors or by ignoring the domain.
When solving trigonometric equations like cos y = 1/2 over a full cycle, the student must find all solutions in the specified interval by considering all quadrants where the function is positive or negative, not just the basic angle.
When solving trigonometric equations over a given interval, you must first convert the interval for x to the corresponding interval for the argument (here, x/6), or else you may include extraneous solutions outside the original range.
When solving trigonometric equations involving tan(kx), factorise instead of dividing by tan(kx) to avoid losing solutions where tan(kx) = 0, and adjust the interval for kx to capture all solutions within the original x-range.
When solving trigonometric equations by factoring, check all factors for solutions within the given interval, and reject impossible values (like sin x outside [−1, 1]).
Common mistake: Missing endpoints or including extraneous roots are common errors.
To solve a trigonometric equation involving sec² and tan, use the identity sec²(θ) = 1 + tan²(θ) to rewrite in terms of tan, factorise, and remember to double the solution range for 2x before halving to find x.
Common mistake: A common mistake is to solve for x directly without adjusting the range for the multiple angle.

Worked example

Solve the equation (3 - 2 cot x) cos x = 0 for 0 < x ≤ pi/2.

Try it first, then show the working

Product is zero, so cos x = 0 or 3 - 2 cot x = 0.
cos x = 0 gives x = pi/2 (= 1.57) in the range 0 < x <= pi/2.
3 - 2 cot x = 0 gives cot x = 3/2, so tan x = 2/3.
x = arctan(2/3) = 0.5880 = 0.588 (3 s.f.), which lies in the range.
Solutions: x = 0.588 and x = pi/2.

Answer: x = 0.588 or x = pi/2 (1.57)

Try one yourself

Hence solve the equation cot y / (cosec y - 1) + cot y / (cosec y + 1) = 4 for 0 < y < 360 degrees.Answer it and get it marked → Solve 5 cos(x/6) + 3 = 0 for -1080 degrees <= x <= 1080 degrees.Answer it and get it marked → Solve tan^2(3x) - 9 tan(3x) = 0 for 0 degrees <= x <= 120 degrees.Answer it and get it marked →

16 exam-style questions on Solving trigonometric equations, each marked to the scheme.

Solving trigonometric equations

Key ideas and common mistakes

Worked example

Try one yourself

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