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Chapter 4: Problem 12

Suppose \(\frac{\pi}{2}<\theta<\pi\) and \(\sin \theta=\frac{3}{4} .\) Evaluate: (a) \(\cos \theta\) (b) \(\tan \theta\)

Short Answer

Expert verified
(a) \(\cos \theta = -\frac{\sqrt{7}}{4}\) (b) \(\tan \theta = \frac{-3\sqrt{7}}{7}\)

Step by step solution

01

Understanding the properties in the second quadrant

Since the angle \(\theta\) lies in the second quadrant, we must recall that in the second quadrant, the value of \(\sin x\) is positive while the values of \(\cos x\) and \(\tan x\) are both negative.
02

Utilize the Pythagorean Identity

In order to find \(\cos \theta\), we will use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). From the given information, we know that \(\sin \theta = \frac{3}{4}\). Plug this into the identity: \(\left(\frac{3}{4}\right)^2 + \cos^2 \theta = 1\)
03

Solve for \(\cos \theta\)

Now, we simply solve for \(\cos \theta\): \[\frac{9}{16} + \cos^2 \theta = 1\] \[\cos^2 \theta = 1 - \frac{9}{16}\] \[\cos^2 \theta = \frac{7}{16}\] Since \(\cos \theta\) is negative in the second quadrant, the solution will be negative: \[\cos \theta = -\frac{\sqrt{7}}{4}\]
04

Utilize the tangent identity

Now we need to find \(\tan \theta\). We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). We have already found both \(\sin \theta\) and \(\cos \theta\), so we can plug those values into the equation: \[\tan \theta = \frac{\frac{3}{4}}{-\frac{\sqrt{7}}{4}}\]
05

Simplify the expression

Now, we simply need to simplify the expression: \[\tan \theta = \frac{3}{- \sqrt{7}}\] Multiplying the numerator and the denominator by \(\sqrt{7}\), we get: \[\tan \theta = \frac{-3\sqrt{7}}{7}\] So, to answer the given question, we found that: (a) \(\cos \theta = -\frac{\sqrt{7}}{4}\) (b) \(\tan \theta = \frac{-3\sqrt{7}}{7}\)

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Most popular questions from this chapter

Show that $$(\cos \theta+\sin \theta)^{2}=1+2 \cos \theta \sin \theta$$ for every number \(\theta\). [Expressions such as \(\cos \theta \sin \theta\) mean \((\cos \theta)(\sin \theta),\) \(\operatorname{not} \cos (\theta \sin \theta) .]\)

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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.] \(\tan \frac{17 \pi}{8}\)

Find exact expressions for the indicated quantities. \(\tan (v-4 \pi)\)

Show that $$\cos (\pi-\theta)=-\cos \theta$$ for every angle \(\theta\).
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