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Chapter 4: Problem 8

Find the four smallest positive numbers \(\theta\) such that \(\tan \theta=-1\).

Short Answer

Expert verified
The four smallest positive angles $\theta$ such that $\tan \theta=-1$ are: 1. \( \theta_1 = 135^\circ = \dfrac{3\pi}{4} \) radians 2. \( \theta_2 = 225^\circ = \dfrac{5\pi}{4} \) radians 3. \( \theta_3 = \dfrac{7\pi}{4} \) radians 4. \( \theta_4 = \dfrac{9\pi}{4} \) radians

Step by step solution

01

Determine the reference angle

The reference angle is the angle formed by the terminal side of an angle in standard position and the x-axis, measured in the positive x-direction. Since \( \tan \theta = -1\), we need to find the reference angle for which \( \tan \theta = 1\). This occurs when \( \theta = 45^\circ \) or \( \theta = \dfrac{\pi}{4} \) radians.
02

Find the angle in the second quadrant

To find the angle with a tangent of -1 in the second quadrant, we subtract the reference angle from \(180^\circ\) or \( \pi \) radians: \( \theta_1 = 180^\circ - 45^\circ = 135^\circ \) or \( \theta_1 = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4} \) radians.
03

Find the angle in the fourth quadrant

To find the angle with a tangent of -1 in the fourth quadrant, we add the reference angle to \(360^\circ - 180^\circ\) or \( 2\pi - \pi \) radians: \( \theta_2 = 360^\circ - 180^\circ + 45^\circ = 225^\circ \) or \( \theta_2 = 2\pi - \pi + \dfrac{\pi}{4} = \dfrac{5\pi}{4} \) radians.
04

Find the next two smallest angles

As mentioned earlier, the tangent function has a period of \( \pi \). Therefore, to find the next two smallest positive angles, we simply add the period to both the angles found in the second and fourth quadrants: \( \theta_3 = \theta_1 + \pi = \dfrac{3\pi}{4} + \pi = \dfrac{7\pi}{4} \) radians and \( \theta_4 = \theta_2 + \pi = \dfrac{5\pi}{4} + \pi = \dfrac{9\pi}{4} \) radians. Now we have found the four smallest positive angles with a tangent value of -1: 1. \( \theta_1 = 135^\circ = \dfrac{3\pi}{4} \) radians 2. \( \theta_2 = 225^\circ = \dfrac{5\pi}{4} \) radians 3. \( \theta_3 = \dfrac{7\pi}{4} \) radians 4. \( \theta_4 = \dfrac{9\pi}{4} \) radians

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

Find the perimeter of an isosceles triangle that has two sides of length 8 and a \(130^{\circ}\) angle between those two sides.

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. \(\sin v\)

Find a formula for the perimeter of an isosceles triangle that has two sides of length \(c\) with angle \(\theta\) between those two sides.

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