/*! 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 10 Find the exact value. (a) \(\co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 10

Find the exact value. (a) \(\cos (5 \pi / 4)\) (b) \(\cos (-11 \pi / 6)\)

Short Answer

Expert verified
(a) \(-\frac{\sqrt{2}}{2}\); (b) \(\frac{\sqrt{3}}{2}\).

Step by step solution

01

Understand the Problem (Part A)

We need to find the exact value of \( \cos \left( \frac{5\pi}{4} \right) \). This angle is expressed in radians and lies in the third quadrant of the unit circle.
02

Convert Angle (Part A)

The angle \( \frac{5\pi}{4} \) is an extension of \( \pi \) by an angle of \( \frac{\pi}{4} \) (i.e., \( \pi + \frac{\pi}{4} \)). Angles in the third quadrant have symmetry with angles in the first quadrant which helps in evaluating trigonometric functions.
03

Determine Reference Angle (Part A)

The reference angle for \( \frac{5\pi}{4} \) is \( \frac{\pi}{4} \). Since this is in the third quadrant, the cosine value is negative, and we find the value for the reference angle.
04

Calculate Cosine Value (Part A)

The exact cosine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \).
05

Understand the Problem (Part B)

We need to find the exact value of \( \cos \left( -\frac{11\pi}{6} \right) \). This angle is negative, meaning it is measured clockwise from the positive x-axis.
06

Convert Angle (Part B)

Add \( 2\pi \) to \( -\frac{11\pi}{6} \) to convert to a positive angle: \( -\frac{11\pi}{6} + 2\pi = \frac{\pi}{6} \). This means \( \cos \left( -\frac{11\pi}{6} \right) = \cos \left( \frac{\pi}{6} \right) \).
07

Calculate Cosine Value (Part B)

The exact cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \). Thus, \( \cos \left( -\frac{11\pi}{6} \right) = \frac{\sqrt{3}}{2} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Unit Circle
The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the trigonometric functions for all angles. Each point on the unit circle corresponds to an angle, measured in radians, and has coordinates
  • The x-coordinate represents the cosine of the angle.
  • The y-coordinate represents the sine of the angle.
For example, an angle of theta corresponds to the point \(( ext{cos} \ heta, ext{sin} \ heta)\) on the circle. The circle makes it easier to visualize angles and understand their corresponding trigonometric values. The unit circle is helpful in recognizing angle cycles and symmetries, particularly between the four quadrants.
Reference Angle
A reference angle is always the acute angle formed between the terminal side of a given angle and the x-axis. It helps simplify the calculation of trigonometric functions.
Reference angles are especially useful when dealing with angles located in the second, third, or fourth quadrants, as they provide a way to relate these angles back to the first quadrant.
  • In the first quadrant, the angle itself is the reference angle.
  • In the second quadrant, subtract the angle from \(\pi\) (or 180 degrees) to find the reference angle.
  • In the third quadrant, subtract \(\pi\) from the angle.
  • In the fourth quadrant, subtract the angle from \(2\pi\) (or 360 degrees).
Reference angles simplify the process of finding trigonometric values by allowing us to use familiar angles from the first quadrant.
Cosine Value Calculation
Finding the cosine value of an angle involves determining the x-coordinate of the corresponding point on the unit circle. For many angles, these values can be memorized or calculated using the reference angle approach.
The cosine function depends on the quadrant of the angle:
  • In the first quadrant, cosine values are positive, as both x and y coordinates are positive.
  • In the second quadrant, cosine values become negative, as the x-coordinates are negative while y-coordinates remain positive.
  • In the third quadrant, both x and y-coordinates are negative, making the cosine negative.
  • In the fourth quadrant, x-coordinates turn positive again, while y-coordinates become negative.
Thus, the cosine value for any angle can be derived from its reference angle and the quadrant it is positioned in. The unit circle makes identifying these cosine values computationally easier by providing a spatial depiction of angle positions.
Quadrants in Trigonometry
The coordinate plane is divided into four quadrants, each playing a vital role in determining the signs of trigonometric functions, including cosine. Understanding these quadrants helps explain the behavior of angles and their respective trigonometric values.
The quadrants are arranged counterclockwise, beginning with the upper right:
  • First Quadrant: Angles between \(0\) and \(\pi/2\) (0 to 90 degrees) have both sine and cosine values positive.
  • Second Quadrant: Angles between \(\pi/2\) and \(\pi\) (90 to 180 degrees) have positive sine and negative cosine values.
  • Third Quadrant: Angles between \(\pi\) and \(3\pi/2\) (180 to 270 degrees) have both sine and cosine values negative.
  • Fourth Quadrant: Angles between \(3\pi/2\) and \(2\pi\) (270 to 360 degrees) have negative sine but positive cosine values.
This understanding of quadrants is significant when predicting the sign of trigonometric values for non-acute angles, making calculations accurate and coherent with the angles' spatial positioning on the unit circle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$37^{\circ} 41^{\prime}$$

Magnetic pole drift The geographic and magnetic north poles have different locations. Currently, the magnetic north pole is drifting westward through \(0.0017\) radian per year, where the angle of drift has its vertex at the center of Earth. If this movement continues, approximately how many years will it take for the magnetic north pole to drift a total of \(5^{\circ}\) ?

An airplane takes off at a \(10^{\circ}\) angle and travels at the rate of \(250 \mathrm{ft} / \mathrm{sec}\). Approximately how long does it take the airplane to reach an altitude of 15,000 feet? 33 Designing a drawbridge

Trigonometric functions are used extensively in the design of industrial robots. Suppose that a robot's shoulder joint is motorized so that the angle \(\theta\) increases at a constant rate of \(\pi / 12\) radian per second from an initial angle of \(\theta=0\). Assume that the elbow joint is always kept straight and that the arm has a constant length of 153 centimeters, as shown in the figure. (a) Assume that \(h=50 \mathrm{~cm}\) when \(\theta=0\). Construct a table that lists the angle \(\theta\) and the height \(h\) of the robotic hand every second while \(0 \leq \theta \leq \pi / 2\). (b) Determine whether or not a constant increase in the angle \(\theta\) produces a constant increase in the height of the hand. (c) Find the total distance that the hand moves.

From a point 15 meters above level ground, a surveyor measures the angle of depression of an object on the ground at \(68^{\circ}\). Approximate the distance from the object to the point on the ground directly beneath the surveyor.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.