Mathematics

Mathematics

Problem

Which of the following gives tan⁡θ\tan\theta in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta for an acute angle θ\theta? a) cos⁡θsin⁡θ\dfrac{\cos\theta}{\sin\theta} b) sin⁡θcos⁡θ\sin\theta\cos\theta c) sin⁡θcos⁡θ\dfrac{\sin\theta}{\cos\theta} d) sin⁡2θ+cos⁡2θ\sin^2\theta+\cos^2\theta
Intuitive explanation

The key insight is that tangent is fundamentally a ratio of the opposite and adjacent sides in a right triangle—which translates directly to the ratio of sine and cosine. Since sin⁡θ\sin\theta gives you the opposite side (normalized by the hypotenuse) and cos⁡θ\cos\theta gives you the adjacent side (also normalized by the hypotenuse), dividing them cancels out the hypotenuse and leaves you with the ratio you need.

Intuitive explanation

The key insight is that tangent is fundamentally a ratio of the opposite and adjacent sides in a right triangle—which translates directly to the ratio of sine and cosine. Since sin⁡θ\sin\theta gives you the opposite side (normalized by the hypotenuse) and cos⁡θ\cos\theta gives you the adjacent side (also normalized by the hypotenuse), dividing them cancels out the hypotenuse and leaves you with the ratio you need.

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