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Chapter 4: Problem 11

Suppose \(\frac{\pi}{2}<\theta<\pi\) and \(\sin \theta=\frac{2}{3}\). Evaluate: (a) \(\cos \theta\) (b) \(\tan \theta\)

Short Answer

Expert verified
(a) \(\cos \theta = -\sqrt{\frac{5}{9}}\) (b) \(\tan \theta = -\frac{2}{\sqrt{5}}\)

Step by step solution

01

Find the value of \(\cos \theta\)

Using the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\), we can solve for \(\cos \theta\): \(1 - \sin^2 \theta = \cos^2 \theta\) We are given that \(\sin \theta=\frac{2}{3}\), so: \(1 - \left(\frac{2}{3}\right)^2 = \cos^2 \theta\) Calculate the result: \(1 - \frac{4}{9} = \frac{9-4}{9} = \frac{5}{9} = \cos^2 \theta\) To find the value of \(\cos \theta\), take the square root: \(\cos \theta = \pm\sqrt{\frac{5}{9}}\) Since \(\theta\) is in the second quadrant, \(\cos \theta\) is negative. Thus, \(\cos \theta = -\sqrt{\frac{5}{9}}\).
02

Find the value of \(\tan \theta\)

We can now find the value of \(\tan \theta\) using the relationship \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Given \(\sin \theta = \frac{2}{3}\) and \(\cos \theta = -\sqrt{\frac{5}{9}}\): \(\tan \theta = \frac{\frac{2}{3}}{-\sqrt{\frac{5}{9}}}\) To simplify, multiply the numerator and denominator by \(3\): \(\tan \theta = \frac{2}{-3\sqrt{\frac{5}{9}}}\) \(\tan \theta = \frac{2}{-3\cdot\frac{\sqrt{5}}{3}}\) The 3's in the denominator will cancel each other: \(\tan \theta = \frac{2}{-\sqrt{5}}\) So, \(\tan \theta = - \frac{2}{\sqrt{5}}\). Now we have both values: (a) \(\cos \theta = -\sqrt{\frac{5}{9}}\) (b) \(\tan \theta = -\frac{2}{\sqrt{5}}\)

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Most popular questions from this chapter

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\tan \left(-\frac{\pi}{8}\right)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\sin (u-6 \pi)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\tan (-\gamma)\)

Show that $$\sin ^{2} \theta=\frac{\tan ^{2} \theta}{1+\tan ^{2} \theta}$$ for all \(\theta\) except odd multiples of \(\frac{\pi}{2}\).

Find exact expressions for the indicated quantities. \(\sin \left(\frac{\pi}{2}-u\right)\)
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