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NSWMaths AdvancedQuick questions

Year 11: Trigonometry

Quick questions on Trigonometric identities for HSC Maths Advanced: the Pythagorean identity sin squared theta plus cos squared theta equals one and its rearrangements, the ratio identity tan theta equals sin theta over cos theta, proving identities by working one side to the other, and simplifying trigonometric expressions, all in degrees

2short 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 proving an identity?
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An identity is an equation that is true for every value of θ\theta for which both sides are defined. That is different from an equation you solve, where you are hunting for the particular angles that make it true. Because an identity is already claimed to be true everywhere, you do not solve it. You demonstrate it, by transforming one side until it becomes the other, with a justified reason at each step.
What is simplifying a trigonometric expression?
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Simplifying is proving without a target: you are handed one expression and asked to reduce it to its shortest equivalent, typically a single ratio such as sin⁡θ\sin\theta or cos⁡θ\cos\theta, or the constant 11. The toolkit is identical. Rewrite any tan⁡\tan as sin⁡θcos⁡θ\dfrac{\sin\theta}{\cos\theta}, put fractions over a common denominator, replace 1−cos⁡2θ1 - \cos^2\theta by sin⁡2θ\sin^2\theta (or 1−sin⁡2θ1 - \sin^2\theta by cos⁡2θ\cos^2\theta), then cancel common factors. The expression usually collapses to one ratio.

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