If sin θ + cos θ = $\frac{\sqrt3}{2}$, and θ is an acute angle, find the value of sin² θ + cos² θ − 2sin θ cos θ.
5/4
We are given \(\sin\theta + \cos\theta = \dfrac{\sqrt{3}}{2}\) and must find \(\sin^2\theta + \cos^2\theta - 2\sin\theta\cos\theta\).
Square the given sum: \((\sin\theta + \cos\theta)^2 = \left(\dfrac{\sqrt{3}}{2}\right)^2 = \dfrac{3}{4}\).
Expanding the left side, \(\sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta = \dfrac{3}{4}\).
Since \(\sin^2\theta + \cos^2\theta = 1\), this gives \(1 + 2\sin\theta\cos\theta = \dfrac{3}{4}\), so \(2\sin\theta\cos\theta = -\dfrac{1}{4}\).
The required expression is \(\sin^2\theta + \cos^2\theta - 2\sin\theta\cos\theta = 1 - 2\sin\theta\cos\theta = 1 - \left(-\dfrac{1}{4}\right) = \dfrac{5}{4}\).
Key concept: \((\sin\theta \pm \cos\theta)^2 = 1 \pm 2\sin\theta\cos\theta\) lets you extract \(2\sin\theta\cos\theta\) from a given sum without finding the angle itself.
Hence the value is 5/4.
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