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Trigonometry and Identities

Quick questions on Solving trigonometric equations explained: O-Level A-Maths

7short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the basic angle?
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The basic angle is the acute angle whose sine, cosine or tangent has the same magnitude as the value you want. Find it from the inverse function applied to the positive value: for sin⁡θ=12\sin\theta = \tfrac{1}{2}, the basic angle is sin⁡−112=30∘\sin^{-1}\tfrac{1}{2} = 30^\circ.
What is equations in a multiple angle?
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If the equation is in 2θ2\theta or 3θ3\theta, first widen the range to match (for 2θ2\theta over 0∘0^\circ to 360∘360^\circ, work in 0∘0^\circ to 720∘720^\circ), solve for the multiple angle, then divide each solution back down. This recovers solutions you would otherwise miss.
What is equations reducible to a quadratic?
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When an equation mixes sin⁡2θ\sin^2\theta and sin⁡θ\sin\theta (or uses an identity to get there), substitute and treat it as a quadratic in the ratio, solve, then solve each resulting simple equation, rejecting any value outside [−1,1][-1, 1].
What is wrong quadrants for the sign?
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Use the sign of the value to pick quadrants; a negative cosine, for instance, lives in the second and third quadrants.
What is q1?
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Solve cos⁡θ=12\cos\theta = \dfrac{1}{2} for 0∘≤θ≤360∘0^\circ \leq \theta \leq 360^\circ. [2 marks]
What is q2?
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Solve tan⁡θ=−1\tan\theta = -1 for 0∘≤θ≤360∘0^\circ \leq \theta \leq 360^\circ. [2 marks]
What is q3?
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Solve sin⁡θ=−12\sin\theta = -\dfrac{1}{2} for 0∘≤θ≤360∘0^\circ \leq \theta \leq 360^\circ. [3 marks]

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