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Chapter 4: Problem 33

(a) Sketch a radius of the unit circle corresponding to an angle \(\theta\) such that \(\cos \theta=\frac{6}{7}\). (b) Sketch another radius, different from the one in part (a), also illustrating \(\cos \theta=\frac{6}{7}\).

Short Answer

Expert verified
For part (a), find angle \(\theta\) using inverse cosine: \(\theta = \arccos{\frac{6}{7}}\). Sketch a radius with angle \(\theta\) in the first quadrant. For part (b), find angle \(\theta'\) in the fourth quadrant: \(\theta' = 2\pi - \theta = 2\pi - \arccos{\frac{6}{7}}\). Sketch a radius with angle \(\theta'\) in the fourth quadrant. Both radii have the same cosine value of \(\frac{6}{7}\).

Step by step solution

01

Determine the angle with the given cosine value

For part (a), we need to find an angle \(\theta\) such that \(\cos \theta = \frac{6}{7}\). Since the cosine function is positive in both the first and second quadrants, we can consider an angle in the first quadrant for this example. Using the inverse cosine function, we can find the angle \(\theta\) as follows: \[ \theta = \arccos{\frac{6}{7}} \] Now we have the angle, we can proceed to sketch the radius.
02

Sketch the radius in part (a)

On the unit circle, the radius length is 1. Starting from the positive x-axis, sketch a radius that makes an angle of \(\theta\) with the x-axis, such that the angle is measured counterclockwise. Label this radius and the angle as part (a).
03

Determine the angle for part (b)

Now we need to find a different angle for part (b) that also has the same cosine value of \(\frac{6}{7}\). Since the cosine function is symmetrical about the y-axis, we can use the symmetry property to determine the angle \(\theta'\) for part (b). Since we used a positive angle in the first quadrant for part (a), use a positive angle in the fourth quadrant for part (b) such that: \[ \theta' = 2\pi - \theta = 2\pi - \arccos{\frac{6}{7}} \]
04

Sketch the radius in part (b)

With the angle \(\theta'\) determined, we can now sketch the radius for part (b) on the unit circle. Starting from the positive x-axis, sketch a radius that makes an angle of \(\theta'\) with the x-axis, such that the angle is measured counterclockwise. Label this radius and the angle as part (b). Now we have successfully sketched two different radii corresponding to different angles, both with the cosine value of \(\frac{6}{7}\).

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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.] \(\tan \frac{13 \pi}{12}\)

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