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Trigonometry and Identities

Quick questions on The R-formula explained: O-Level A-Maths

6short 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 are solving equations?
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To solve asin⁡θ+bcos⁡θ=ka\sin\theta + b\cos\theta = k, rewrite the left as Rsin⁡(θ+α)=kR\sin(\theta + \alpha) = k, so sin⁡(θ+α)=kR\sin(\theta + \alpha) = \dfrac{k}{R}, an equation in a single ratio that you solve in the usual way.
What is wrong matching for a cosine form?
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For Rcos⁡(θ+α)R\cos(\theta + \alpha) the cosine coefficient matches Rcos⁡αR\cos\alpha and the sine coefficient matches Rsin⁡αR\sin\alpha; align with the chosen form carefully.
What is quadrant errors in α\alpha?
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With the standard positive RR and acute α\alpha, α=tan⁡−1(b/a)\alpha = \tan^{-1}(b/a) is acute; do not add a sign.
What is q1?
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Express sin⁡θ+cos⁡θ\sin\theta + \cos\theta in the form Rsin⁡(θ+α)R\sin(\theta + \alpha). [3 marks]
What is q2?
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State the maximum value of 6sin⁡θ+8cos⁡θ6\sin\theta + 8\cos\theta. [2 marks]
What is q3?
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State the minimum value of 5+3sin⁡θ+4cos⁡θ5 + 3\sin\theta + 4\cos\theta. [2 marks]

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