Mathematics

Mathematics

Problem

If cos⁡θ=−32\cos\theta=-\dfrac{\sqrt3}{2} and θ\theta lies in Quadrant III, the value of sin⁡θ\sin\theta is a) −32-\dfrac{\sqrt3}{2} b) 12\dfrac{1}{2} c) −12-\dfrac{1}{2} d) 32\dfrac{\sqrt3}{2}
Intuitive explanation

The key is recognizing that cosine and sine are linked through the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Once you substitute the given cosine value and solve for sin⁡2θ\sin^2\theta, you'll get two possible answers—but the quadrant tells you which sign is correct. Since sine is negative in Quadrant III, you'll pick the negative solution.

Intuitive explanation

The key is recognizing that cosine and sine are linked through the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Once you substitute the given cosine value and solve for sin⁡2θ\sin^2\theta, you'll get two possible answers—but the quadrant tells you which sign is correct. Since sine is negative in Quadrant III, you'll pick the negative solution.

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