/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Evaluate (if possible) the sine,... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 19

Evaluate (if possible) the sine, cosine, and tangent at the real number. $$t=\frac{11 \pi}{6}$$

Short Answer

Expert verified
The sine of \( \frac{11 \pi }{6} \) is \( \frac{1}{2} \). The cosine of \( \frac{11 \pi }{6} \) is \( \frac{\sqrt{3}}{2} \). The tangent of \( \frac{11 \pi }{6} \) is \( \frac{\sqrt{3}}{3} \).

Step by step solution

01

Understand the Problem

The problem is asking for the sine, cosine, and tangent of a specific angle. This angle is given as a fraction of \( \pi \), which means it's a direction on the unit circle. The sine is the y-coordinate of the point on the unit circle. The cosine is the x-coordinate. The tangent is the sine divided by the cosine, which is the ratio of the y-coordinate to the x-coordinate.
02

Identify the Angle on the Unit Circle

The angle given is \( \frac{11 \pi }{6} \). Convert this to an angle in a unit circle. This angle is equivalent to \( \frac{\pi }{6} \) because \( \frac{11 \pi }{6} = 2\pi + \frac{\pi }{6} \). The extra 2\(\pi\) just means we've gone around the unit circle twice, and doesn't change the resulting point on the circle. This is because the unit circle repeats every 2\(\pi\) radians.
03

Find the Cosine of the Angle

The cosine of \( \frac{\pi }{6} \) is \( \frac{\sqrt{3}}{2} \). This results from the definition of cosine as the x-coordinate of the point on the unit circle, and knowing that the point corresponding to \( \frac{\pi }{6} \) is \( \left(\frac{\sqrt{3}}{2},\frac{1}{2}\right) \).
04

Find the Sine of the Angle

The sine of \( \frac{\pi }{6} \) is \( \frac{1}{2} \). This results from the definition of sine as the y-coordinate of the point on the unit circle, and knowing that the point corresponding to \( \frac{\pi }{6} \) is \( \left(\frac{\sqrt{3}}{2},\frac{1}{2}\right) \).
05

Find the Tangent of the Angle

The tangent of \( \frac{\pi }{6} \) is \( \frac{sine}{cosine} = \frac{1/2}{\sqrt{3}/2} = \frac{\sqrt{3}}{3} \). This results from the definition of tangent as the ratio of the y-coordinate to the x-coordinate on the unit circle.

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