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Chapter 5: Problem 21

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\sin u=-3 / 5, \quad 3 \pi / 2

Short Answer

Expert verified
\(\sin 2u = -24/25\), \(\cos 2u = 7/25\), and \(\tan 2u = -24/7\).

Step by step solution

01

Find Value of Cosine

Since the sine ratio is given and we know the sine ratio is \(-3 / 5\), we can find the value of cosine using the Pythagorean identity which gives \(\cos u = \sqrt{1 - \sin^2 u }\), substituting the value of sine we get, \( \cos u = \sqrt{1 - (-3/5)^2} = -4 / 5\). The negative sign is due to the fact that cosine in the 4th quadrant is negative.
02

Apply the Formulas

Now that we have the values for both \(\sin u\) and \(\cos u\), we can use the double-angle identities to find the desired values: \For \(\sin 2u\), \(\sin 2u = 2 \sin u \cos u\), substituting values we get, \(\sin 2u = -2 * (-3/5) * (-4/5) = - 24 / 25\). \For \(\cos 2u\), \(\cos 2u = \cos^{2} u - \sin^{2} u\), substituting values we get, \(\cos 2u = (-4/5)^{2} - (-3/5)^{2} = 7 / 25\). \For \(\tan 2u\), \(\tan 2u = \sin 2u / \cos 2u\), substituting values we get, \(\tan 2u = -24/25 / 7/25 = -24/7\).
03

Presenting the Final Result

The exact values of the double angle functions are then: \(\sin 2u = -24/25\), \(\cos 2u = 7/25\), and \(\tan 2u = -24/7\).

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