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Chapter 5: Problem 22

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\cos u=-4 / 5, \quad \pi / 2

Short Answer

Expert verified
The exact values for the equations are \(\sin 2u = -24/25\), \(\cos 2u = 7/25\) and \(\tan 2u = -24/7\).

Step by step solution

01

Compute the value of \(\sin u\)

In the second quadrant, since \(\cos u = -4/5\), we can use the identity \(\sin^2 u + \cos^2 u = 1\) to calculate \(\sin u\). Thus we have \(\sin u = \sqrt{1 - \cos^2 u} = \sqrt{1 - (-4/5)^2} = \sqrt{1 - 16/25} = \sqrt{9/25} = 3/5\).
02

Apply the double-angle formulas

The double angle formulas are \(\sin 2u = 2 \sin u \cos u\), \(\cos 2u = \cos^2 u - \sin^2 u\) or \(\cos 2u = 2\cos^2 u - 1\) or \(\cos 2u = 1 - 2\sin^2 u\), and \(\tan 2u = \sin 2u / \cos 2u\).
03

Compute \(\sin 2u\)

Using the formula for \(\sin\) in a double angle, we get \(\sin 2u = 2 \sin u \cos u = 2 * 3/5 * -4/5 = -24/25\).
04

Compute \(\cos 2u\)

Using the formula for \(\cos\) in a double angle, we get \(\cos 2u = \cos^2 u - \sin^2 u = (-4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25\).
05

Compute \(\tan 2u\)

Using the formula for \(\tan\) in a double angle, we get \(\tan 2u = \sin 2u / \cos 2u = -24/25 / 7/25 = -24/7\).

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