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Chapter 4: Problem 48

Find the lengths of both circular arcs on the unit circle connecting the point \(\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\) and the endpoint of the radius corresponding to \(50^{\circ} .\)

Short Answer

Expert verified
The lengths of the circular arcs on the unit circle connecting the given point and the endpoint of the radius corresponding to \(50^{\circ}\) are \(\frac{7\pi}{18}\) and \(\frac{29\pi}{18}\).

Step by step solution

01

Find the central angle of the given point

First, we need to find the angle corresponding to the point \(\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\). Using the atan2 function (inverse tangent of the quotient of the coordinates), the central angle in radians is \[ \theta_1 = atan2 \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) = -\frac{\pi}{6}. \] The negative sign indicates that the point is below the x-axis, so the angle should be measured clockwise from the positive x-axis.
02

Find the central angle of the point corresponding to \(50^{\circ}\)

In order to find this central angle, we need to convert the angle in degrees to radians: \[ \theta_2 = 50^{\circ} \cdot \frac{\pi}{180^{\circ}}=\frac{5\pi}{18}. \]
03

Calculate the difference of angles

With both angles in radians, we can now calculate the difference between the angles: \[ \Delta \theta_1 = \theta_2 - \theta_1 = \frac{5\pi}{18} - \left(-\frac{\pi}{6}\right) = \frac{7\pi}{18}. \]
04

Calculate the complementary angle of the difference

Since there's another arc on the unit circle connecting the two points, we need to find the complementary angle, i.e., the angle formed by the longer arc. The total angle on a circle is equal to \(2\pi\): \[ \Delta \theta_2 = 2\pi - \Delta \theta_1 = 2\pi - \frac{7\pi}{18} = \frac{29\pi}{18}. \]
05

Calculate the lengths of the arcs

Finally, we can use the relationship that the arc length is equal to the product of the angle in radians and the radius of the circle. In this case, because we are working with the unit circle (radius = 1), the arc lengths are equal to the angles in radians: \[ \text{Arc Length}_1 = \Delta \theta_1 = \frac{7\pi}{18} \] and \[ \text{Arc Length}_2 = \Delta \theta_2 = \frac{29\pi}{18}. \] Hence, the lengths of the circular arcs on the unit circle connecting the given point and the endpoint of the radius corresponding to \(50^{\circ}\) are \(\frac{7\pi}{18}\) and \(\frac{29\pi}{18}\).

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

Find the smallest positive number \(x\) such that $$ \sin (x+\pi)-\sin x=1 $$

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.] \(\tan \left(-\frac{5 \pi}{12}\right)\)

Simplify the expression $$ (\tan \theta)\left(\frac{1}{1-\cos \theta}-\frac{1}{1+\cos \theta}\right) . $$

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