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Chapter 4: Problem 14

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=\frac{1}{5} .\) Evaluate: (a) \(\sin \theta\) (b) \(\tan \theta\)

Short Answer

Expert verified
(a) \(\sin \theta = -\frac{2\sqrt{6}}{5}\) (b) \(\tan \theta = -2\sqrt{6}\)

Step by step solution

01

Recall the Pythagorean identity

Recall the Pythagorean identity for trigonometric functions: \(\sin^2 \theta + \cos^2 \theta = 1\). Since we know the value of \(\cos \theta\), we can substitute this into the equation and then solve for \(\sin \theta\).
02

Determine the sine value of 胃

Given \(\cos \theta =\frac{1}{5}\), let's substitute this value into the Pythagorean identity: \(\sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1\) Now, solve for \(\sin^2 \theta\): \(\sin^2 \theta = 1 - \left(\frac{1}{5}\right)^2\) \(\sin^2 \theta = 1 - \frac{1}{25}\) \(\sin^2 \theta = \frac{24}{25}\) Take the square root of both sides, keeping in mind that since \(-\frac{\pi}{2}<\theta<0\), the sine value will be negative: \(\sin \theta = -\sqrt{\frac{24}{25}}\) So, \(\sin \theta = -\frac{2\sqrt{6}}{5}\).
03

Determine the tangent value of 胃

Recall the definition of the tangent function: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). We have the values for \(\sin \theta\) and \(\cos \theta\), so let's substitute them into the equation: \(\tan \theta = \frac{-\frac{2\sqrt{6}}{5}}{\frac{1}{5}}\) Now, simplify the expression by multiplying the numerator and denominator by 5: \(\tan \theta = \frac{-2\sqrt{6}}{1}\) So, \(\tan \theta = -2\sqrt{6}\). In conclusion, the solutions to the exercise are: (a) \(\sin \theta = -\frac{2\sqrt{6}}{5}\) (b) \(\tan \theta = -2\sqrt{6}\)

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Most popular questions from this chapter

Assume the surface of the earth is a sphere with radius 3963 miles. The latitude of \(a\) point \(P\) on the earth's surface is the angle between the line from the center of the earth to \(P\) and the line from the center of the earth to the point on the equator closest to \(P\), as shown below for latitude \(40^{\circ} .\) Bow fast is Cleveland moving due to the daily rotation of the earth about its axis?

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.] \(\tan \frac{9 \pi}{8}\)

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.] \(\tan \frac{25 \pi}{12}\)

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. \(\cos (u+4 \pi)\)

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