/*! 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 76 Find the exact value of each exp... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 76

Find the exact value of each expression. Do not use a calculator. $$ \sec \frac{11 \pi}{4} $$

Short Answer

Expert verified
-\sqrt{2}

Step by step solution

01

Reduce the angle

First, reduce the angle \( \frac{11 \pi}{4} \) to a principal angle between \( 0 \) and \( 2\pi \). To do this, subtract multiples of \( 2\pi \) from \( \frac{11\pi}{4} \). Since \( 2\pi = \frac{8\pi}{4} \), we have \( \frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4} \). Hence, \( \frac{11\pi}{4} = \frac{3\pi}{4} \) in the principal range.
02

Identify the reference angle

The angle \( \frac{3\pi}{4} \) is in the second quadrant. To find the reference angle, subtract it from \( \pi \). The reference angle is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).
03

Determine the cosine of the reference angle

The reference angle is \( \frac{\pi}{4} \). In the second quadrant, cosine is negative. Thus, \( \cos \frac{3\pi}{4} = - \cos \frac{\pi}{4} \). Since \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), it follows that \( \cos \frac{3\pi}{4} = -\frac{1}{\sqrt{2}} \).
04

Find the secant of the angle

Secant is the reciprocal of cosine. Therefore, \( \sec \frac{3\pi}{4} = \frac{1}{\cos \frac{3\pi}{4}} = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

unit circle
The unit circle is a circle centered at the origin (0,0) of a coordinate plane with a radius of 1. It is a powerful tool in trigonometry because it helps to define the sine, cosine, and tangent functions for all angles. < br > Every point on the unit circle corresponds to an \(x, y\) value, which can be interpreted as the coordinates \(\cos(\theta), \sin(\theta)\) for a given angle \(\theta\). < br > Some key points to remember about the unit circle:
  • Each angle in the unit circle is measured in radians.
  • The circle makes it easy to find the values of trigonometric functions at various angles.
  • The values of sine and cosine range between -1 and 1, while tangent can take any real number.
Using the unit circle simplifies working with trigonometric functions and provides a visual understanding of how these functions behave at different angles.

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Most popular questions from this chapter

Convert each angle in radians to degrees. Express your answer in decimal form, rounded to two decimal places. 3.14

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places.\(125^{\circ}\)

Name the quadrant in which the angle \(\theta\) lies. $$ \cos \theta>0, \quad \cot \theta<0 $$

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Geometry The hypotenuse of a right triangle has a length of 8 inches. If one angle is \(35^{\circ}\), find the length of each leg.
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