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Chapter 5: Problem 20

Find the four smallest positive numbers \(\theta\) such that \(\cos \theta=\frac{1}{2}\)

Short Answer

Expert verified
The four smallest positive numbers 胃 such that \(\cos{\theta} = \frac{1}{2}\) are: 1. \(\theta_1 = \frac{\pi}{3}\) 2. \(\theta_2 = \frac{5\pi}{3}\) 3. \(\theta_3 = \frac{7\pi}{3}\) 4. \(\theta_4 = \frac{11\pi}{3}\)

Step by step solution

01

Identify Quadrants With Cosine Equal to 1/2

Since cosine is positive in the first and fourth quadrants, we'll first find the angles in those quadrants whose cosine values equal 1/2.
02

Use Reference Angles

The reference angle for which \(\cos{\theta} = \frac{1}{2}\) is 60 degrees or \(\frac{\pi}{3}\) radians in the first and fourth quadrants. To find the corresponding angles in both the first and fourth quadrants in radians, we can use the formulas: 1. 胃 in first quadrant: \(\theta = \frac{\pi}{3}\) 2. 胃 in fourth quadrant: \(\theta = 2\pi - \frac{\pi}{3}\)
03

Calculate The Two Smallest Positive Angles

Calculate the angles in the first and fourth quadrants using the formulas from Step 2: 1. 胃 in first quadrant: \(\theta_1 = \frac{\pi}{3} = \frac{\pi}{3} \text{ radians}\) 2. 胃 in fourth quadrant: \(\theta_2 = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \text{ radians}\) These are the two smallest positive angles that satisfy \(\cos{\theta} = \frac{1}{2}\).
04

Use Periodicity of Cosine Function

As the cosine function has a period of \(2\pi\), we can obtain the next smallest solutions by adding multiples of \(2\pi\) to the angles found in Step 3: 3. Third smallest angle: \(\theta_3 = \theta_1 + 2\pi = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3} \text{ radians}\) 4. Fourth smallest angle: \(\theta_4 = \theta_2 + 2\pi = \frac{5\pi}{3} + 2\pi = \frac{11\pi}{3} \text{ radians}\)
05

Write the Solution

The four smallest positive numbers 胃 such that \(\cos{\theta} = \frac{1}{2}\) are: 1. \(\theta_1 = \frac{\pi}{3}\) 2. \(\theta_2 = \frac{5\pi}{3}\) 3. \(\theta_3 = \frac{7\pi}{3}\) 4. \(\theta_4 = \frac{11\pi}{3}\)

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Most popular questions from this chapter

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 4 and 5 in Section 6.3.] $$ \tan \left(-\frac{3 \pi}{8}\right) $$

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

Find all numbers \(t\) such that $$ \cos ^{-1} t=\sin ^{-1} t $$.

Show that $$ \tan ^{-1} \frac{1}{t}=\frac{t}{|t|} \frac{\pi}{2}-\tan ^{-1} t $$ for all \(t \neq 0\)

Evaluate \(\cos \left(\tan ^{-1}(-4)\right)\).
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