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Chapter 5: Problem 23

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\tan u=3 / 5, \quad 0

Short Answer

Expert verified
The exact values are \(\sin 2u = \frac{24}{25}\), \(\cos 2u = \frac{7}{25}\) and \(tan 2u = \frac{24}{7}\).

Step by step solution

01

Convert given tan(u) to sin(u) and cos(u)

Since sin(u) = opposite / hypotenuse and cos(u) = adjacent / hypotenuse, by arranging Pythagorean identity we can find that opposite = 3, adjacent = 4 and hypotenuse = 5. Thus, \(\sin u = \frac{3}{5}\) and \(\cos u = \frac{4}{5}\). Note that \(sin^2 u + cos^2 u = 1\), which holds correct here.
02

Apply double-angle formulas for sin(2u) and cos(2u)

Using the double-angle formulas, we know that \(\sin 2u = 2 \sin u \cos u\) and \(\cos 2u = \cos^2 u - \sin^2 u\). Substituting the values of sin(u) and cos(u) that we calculated in Step 1, we obtain \(\sin 2u = 2 * \frac{3}{5} * \frac{4}{5} = \frac{24}{25}\) and \(\cos 2u = (\frac{4}{5})^2 - (\frac{3}{5})^2 = \frac{7}{25}\).
03

Apply the formula for tan(2u)

Lastly, use the double-angle formula for tan(2u), which is \(tan(2u) = \frac{sin(2u)}{cos(2u)}\). Substitute the values of sin(2u) and cos(2u) calculated in the previous step to get \(tan(2u) = \frac{\frac{24}{25}}{\frac{7}{25}} = \frac{24}{7}\).

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Most popular questions from this chapter

(a) determine the quadrant in which \(u / 2\) lies, and (b) find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=5 / 13, \quad \pi / 2

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