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Chapter 5: Problem 24

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

Short Answer

Expert verified
The exact values of the trigonometric functions are: \(\sin 2u = \sqrt{3}/2, \cos 2u = -1/2, \tan 2u = -\sqrt{3}\).

Step by step solution

01

Identify the Sign of tan \(u\)

The secant function \(sec u = -2\) is negative, and the interval \(\pi < u < 3\pi / 2\) lies in the third quadrant where tan is positive. So, \(tanu\) should be positive.
02

Calculate \(tanu\)

Use the relationship between sec and tan in the Pythagorean identity: \(sec^2u = 1+tan^2u\). By rearranging the equation, we find that \(tan^2u = sec^2u - 1\). Substituting the given \(secu = -2\), it becomes \(tan^2u = (-2)^2 -1 = 4-1 = 3\), thus \(tanu = \sqrt{3}\).
03

Calculate \(sin 2u, cos 2u\) and \(tan 2u\)

Using the double angle formulas: \[sin 2u = 2sinu cosu = 2tanu/(1+tan^2u) = 2*sqrt{3}/(1+3) = sqrt{3}/2\]. \[ cos 2u = 1- 2sin^2u = 1- 2(tan^2u / (1+ tan^2u)) = 1- 2(3/4) = -1/2\]. \( tan 2u = sin 2u / cos 2u = (sqrt{3}/2)/(-1/2) = -sqrt{3}\].

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Most popular questions from this chapter

Write the trigonometric expression as an algebraic expression. $$\sin (\arctan 2 x-\arccos x)$$

Determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \(u, \sin 2 u=-2 \sin u \cos u\).

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

Verify the identity. $$\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$
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