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# What is the exact value of the expressions using the double-angle identity?

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Use a double-angle identity to find the exact value of each expression.

1. sin 120°

2. tan 60°

3. cos 4π/3

4. sin 5π/3

Apr 17, 2019

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$$\sin\left(120^{\circ}\right)\\ = 2\sin\left(60^{\circ}\right)\cos\left(60^{\circ}\right)\\ =2\left(\dfrac{\sqrt3}{2}\right)\left(\dfrac{1}{2}\right)\\ =\dfrac{\sqrt3}2$$

$$\tan\left(60^{\circ}\right)\\ =\dfrac{2\tan\left(30^{\circ}\right)}{1-\tan^2\left(30^{\circ}\right)}\\ =\dfrac{\dfrac{2}{\sqrt3}}{1-\dfrac{1}{3}}\\ =\dfrac{\dfrac{2}{\sqrt3}}{\dfrac{2}{3}}\\ =\dfrac{3}{\sqrt3}\\ =\sqrt3$$

$$\cos\left(\dfrac{4\pi}{3}\right)\\ = 2\cos^2\left(\dfrac{2\pi}3{}\right) - 1\\ = 2\left(2\cos^2\left(\dfrac{\pi}{3}\right)-1\right)^2-1\\ =2\left(2\left(\dfrac{1}{2}\right)^2-1\right)^2 - 1\\ =2\left(-\dfrac{1}{2}\right)^2 - 1\\ =-\dfrac{1}{2}$$

$$\sin\left(\dfrac{5\pi}{3}\right)\\ = 2\sin\left(\dfrac{5\pi}{6}\right)\cos\left(\dfrac{5\pi}{6}\right)\\ = 2\left(\dfrac{1}{2}\right)\left(\dfrac{-\sqrt3}{2}\right)\\ =\dfrac{-\sqrt3}2$$

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Apr 18, 2019