Two ordered triples are proposed to represent for some angle : and .

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Two ordered triples are proposed to represent (sin⁡φ,cos⁡φ,tan⁡φ)(\sin\varphi,\cos\varphi,\tan\varphi) for some angle φ\varphi: T1=(1213,513,125)T_{1}=\bigl(\dfrac{12}{13},\dfrac{5}{13},\dfrac{12}{5}\bigr) and T2=(−513,1213,−125)T_{2}=\bigl(-\dfrac{5}{13},\dfrac{12}{13},-\dfrac{12}{5}\bigr). (a) For each triple, verify whether the first two components satisfy sin⁡2φ+cos⁡2φ=1\sin^{2}\varphi+\cos^{2}\varphi=1." (b) For each triple, check whether the third component equals the quotient of the first two components. (c) Based on your results in parts (i) and (ii), determine whether each triple can represent (sin⁡φ,cos⁡φ,tan⁡φ)(\sin\varphi,\cos\varphi,\tan\varphi) for some real φ\varphi. If it can, state all possible quadrant(s) for φ\varphi; if it cannot, give a clear reason."

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