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Chapter 4: Problem 29

Find the lengths of both circular arcs of the unit circle connecting the point \(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)\) and the point whose radius corresponds to 1 radian.

Short Answer

Expert verified
The lengths of both circular arcs connecting the point \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\) and the point whose radius corresponds to 1 radian are: Arc 1: \(\frac{\pi}{4} + 1 \, \text{radians}\) Arc 2: \(1 - \frac{\pi}{4} \, \text{radians}\)

Step by step solution

01

Find the angle \(\theta\) corresponding to the point

To find the angle for the point \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\), we need to find the inverse tangent of the coordinates. The angle can be determined using the following formula: \[\theta = \tan^{-1}\left( \frac{y}{x} \right)\] Plugging in the coordinates for x and y: \[\theta = \tan^{-1}\left( \frac{- \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\right) = \tan^{-1}(-1)\] After evaluating, we get: \[\theta = - \frac{\pi}{4}\]
02

Determine the angles corresponding to both arcs

We know that \(\theta = - \frac{\pi}{4}\) radians. The two arcs connecting the given point and the point corresponding to 1 radian will have angles differing by 1 radian. Let's find the two angles: Angle for arc 1: \(\theta_1 = - \frac{\pi}{4} - 1\) Angle for arc 2: \(\theta_2 = - \frac{\pi}{4} + 1\)
03

Calculate the lengths of both circular arcs

We know that the arc length is given by the formula: Arc length = radius * angle in radians Since the radius of the unit circle is 1, we can calculate the lengths of both arcs using the angles \(\theta_1\) and \(\theta_2\): Length of arc 1 = \(|\theta_1| = |- \frac{\pi}{4} - 1| = \frac{\pi}{4} + 1\) Length of arc 2 = \(|\theta_2| = |- \frac{\pi}{4} + 1| = 1 - \frac{\pi}{4}\) So, the lengths of both circular arcs connecting the point \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\) and the point whose radius corresponds to 1 radian are: Arc 1: \(\frac{\pi}{4} + 1 \, \text{radians}\) Arc 2: \(1 - \frac{\pi}{4} \, \text{radians}\)

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Most popular questions from this chapter

For Exercises \(29-34,\) assume the surface of the earth is a sphere with radius 3963 miles. The latitude of \(a\) point \(P\) on the earth's surface is the angle between the line from the center of the earth to \(P\) and the line from the center of the earth to the point on the equator closest to \(P\), as shown below for latitude \(40^{\circ} .\) Dallas has latitude \(32.8^{\circ}\) north. Find the radius of the circle formed by the points with the same latitude as Dallas.

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

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Given that $$\cos 15^{\circ}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ Find exact expressions for the indicated quantities. [These values for \(\cos 15^{\circ}\) and \(\sin 22.5^{\circ}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sec 15^{\circ}\)

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