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Chapter 4: Problem 41

(a) Sketch a radius of the unit circle corresponding to an angle \(\theta\) such that \(\tan \theta=\frac{1}{7}\). (b) Sketch another radius, different from the one in part (a), also illustrating \(\tan \theta=\frac{1}{7}\).

Short Answer

Expert verified
In a unit circle, sketch two radii corresponding to angles \(\theta_1\) and \(\theta_2\), where \(\tan \theta_1 = \tan \theta_2 = \frac{1}{7}\). \(\theta_1\) is in the first quadrant with coordinates \(\left(\frac{1}{\sqrt{1+(1/7)^2}}, \frac{1/7}{\sqrt{1+(1/7)^2}}\right)\), while \(\theta_2\) is in the third quadrant with coordinates \(\left(\frac{-1}{\sqrt{1+(-1/7)^2}}, \frac{-1/7}{\sqrt{1+(-1/7)^2}}\right)\).

Step by step solution

01

Draw a unit circle

Draw a unit circle with its center at the origin (0,0) and radius of 1.
02

Definition of tangent

In a unit circle, the tangent of an angle 胃 can be defined as \(\tan 胃 = \frac{\sin 胃}{\cos 胃}\).
03

Locate the points corresponding to \(\tan \theta=\frac{1}{7}\)

Since the tangent function is positive in the first and third quadrants, we are looking for a point in those quadrants such that its angle 胃 satisfies the condition \(\tan 胃 = \frac{1}{7}\). Let's call these angles \(\theta_1\) and \(\theta_2\), where \(\theta_1\) is in the first quadrant and \(\theta_2\) is in the third quadrant.
04

Express sine and cosine in terms of tangent

We know that \(\sin^2 胃 + \cos^2 胃 = 1\). Since \(\tan 胃 = \frac{\sin 胃}{\cos 胃}\), we can rewrite \(\sin 胃\) and \(\cos 胃\) in terms of tangent: \(\sin 胃 = \frac{\tan 胃}{\sqrt{1 + \tan^2 胃}}\) and \(\cos 胃 = \frac{1}{\sqrt{1 + \tan^2 胃}}\) This allows us to find the coordinates of the points corresponding to \(\theta_1\) and \(\theta_2\).
05

Find the coordinates of the points

Since \(\tan \theta = \frac{1}{7}\), we have: For \(\theta_1\): \(\sin \theta_1 = \frac{1/7}{\sqrt{1 + (1/7)^2}}\) \(\cos \theta_1 = \frac{1}{\sqrt{1 + (1/7)^2}}\) For \(\theta_2\): \(\sin \theta_2 = \frac{-1/7}{\sqrt{1 + (-1/7)^2}}\) \(\cos \theta_2 = \frac{-1}{\sqrt{1 + (-1/7)^2}}\)
06

Sketch the radii

Plot the coordinates we found for \(\theta_1\) and \(\theta_2\) on the unit circle and draw the radii connecting the origin to those points. These radii represent the angles \(\theta_1\) and \(\theta_2\), respectively, for which \(\tan \theta=\frac{1}{7}\).

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