Mathematics

Mathematics

Problem

If sin⁡θ=35\sin\theta=\dfrac35 and 0∘<θ<90∘0^\circ<\theta<90^\circ, find tan⁡(90∘−θ)\tan(90^\circ-\theta). a) 34\dfrac34 b) 53\dfrac53 c) 43\dfrac43 d) 45\dfrac45
Intuitive explanation

You're working with complementary angles and their trig ratios. The key insight is that tan⁡(90°−θ)=cot⁡θ\tan(90° - \theta) = \cot\theta, which flips sine and cosine in the ratio. Since you know sin⁡θ\sin\theta, use the Pythagorean identity to find cos⁡θ\cos\theta, then form the cotangent ratio—the reciprocal of tangent.

Intuitive explanation

You're working with complementary angles and their trig ratios. The key insight is that tan⁡(90°−θ)=cot⁡θ\tan(90° - \theta) = \cot\theta, which flips sine and cosine in the ratio. Since you know sin⁡θ\sin\theta, use the Pythagorean identity to find cos⁡θ\cos\theta, then form the cotangent ratio—the reciprocal of tangent.

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