For any acute angle

MATHEMATICS · Multi-part · CBSE · Mathematics · Trigonometric Functions · Trigonometric Functions

Problem

For any acute angle θ\theta, consider the trigonometric ratios sin⁡θ\sin\theta, cos⁡θ\cos\theta and tan⁡θ\tan\theta. Answer the following without using numerical tables. (i) Prove that for θ\theta in (45∘,90∘)\left(45^{\circ},90^{\circ}\right), the inequality sin⁡θ>cos⁡θ\sin\theta>\cos\theta holds. (ii) Using the result of part (i), state whether the quantity sin⁡68∘−cos⁡68∘\sin68^{\circ}-\cos68^{\circ} is positive, negative or zero. Give reason. (iii) Deduce, from part (i), the range of values of θ\theta (in degrees) within (0∘,90∘)(0^{\circ},90^{\circ}) for which tan⁡θ>1\tan\theta>1. Justify each step clearly.

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