/*! 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 33 Evaluate the trigonometric funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 33

Evaluate the trigonometric function using its period as an aid. $$\cos \frac{7 \pi}{3}$$

Short Answer

Expert verified
\(\cos \frac{7 \pi}{3} = \frac{1}{2}\)

Step by step solution

01

Expressing in Periodic Terms

The task is to evaluate \(\cos \frac{7 \pi}{3}\). We can use the periodicity of the cosine function, \(2\pi\), to express \(\frac{7\pi}{3}\) in terms of the period. We have to find an integer n such that \(\frac{7\pi}{3} - 2\pi n\) falls within the range of \([0, 2\pi]\). In this case, if we take n = 1, we get \(\frac{7\pi}{3} - 2\pi = \frac{\pi}{3}\), which falls within the range.
02

Evaluate the Cosine Function

Now, evaluate the cosine function of the resulting angle: \(\cos(\frac{\pi}{3})\). Using the data of cosine function from the unit circle, the value of \(\cos(\frac{\pi}{3})\) is equal to \(\frac{1}{2}\).

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.

Cosine Function
The cosine function, often denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It is primarily used to determine the horizontal component of an angle on the unit circle.
It arises in contexts where we are dealing with right triangles, periodic phenomena, and wave functions, amongst other areas.
  • Definition: The cosine of an angle \( \theta \) is the x-coordinate of the point on the unit circle that corresponds to \( \theta \).
  • Range and Domain: The cosine function's domain includes all real numbers, \( \theta \in \mathbb{R} \), and its range is between -1 and 1, inclusive.
When interpreting the cosine function graphically, the function makes a beautiful wave鈥攁lways returning to the same values at regular intervals, due to its periodic nature. Understanding the cosine function is essential, as it connects to many real-world applications like physics, engineering, and even music.
Periodicity
Periodicity is a crucial concept when working with trigonometric functions like cosine. It refers to the characteristic of a function to repeat its values at regular intervals.
For the cosine function, this period is \(2\pi\). Applying the periodicity principle can simplify evaluations significantly:
  • Function Repeat: This means that \( \cos(\theta) = \cos(\theta + 2n\pi) \) for any integer \( n \).
  • Utility: This characteristic helps us evaluate trigonometric functions for larger angles by reducing them to an equivalent angle within one cycle \([0, 2\pi)\).
This feature of the cosine function is particularly useful in solving problems which involve angles greater than \(2\pi\), by essentially "resetting" the cycle.
Unit Circle
The unit circle is a fundamental tool in trigonometry, depicting a circle with a radius of one centered at the origin of a coordinate plane. This circle allows us to define trigonometric functions for all real numbers based on the x and y coordinates of points on the circle.
  • Coordinates: Any given angle \( \theta \) corresponds to a point \((\cos \theta, \sin \theta)\) on the unit circle.
  • Symmetry: The unit circle is symmetric about the x-axis, y-axis, and origin, which helps in understanding the periodic behavior of trigonometric functions.
The unit circle is indispensable when learning about angle measures, as it provides a consistent reference for calculating trigonometric values such as sine and cosine.
In the given example, the cosine value \( \cos(\frac{\pi}{3}) \) corresponds to the x-coordinate on the unit circle.
Angle Reduction
Angle reduction simplifies the evaluation of trigonometric functions by using an equivalent angle within the principal range, typically \([0, 2\pi)\) for radians. This is done by either adding or subtracting multiples of \( \pi \) or \(2\pi\) to reduce the original angle.
  • Using Period: For example, to evaluate \( \cos\left(\frac{7\pi}{3}\right) \), we find a reduced angle by subtracting \(2\pi\), resulting in \( \frac{\pi}{3} \).
  • Practical Benefits: This method helps to avoid dealing with excessively large numbers while maintaining accuracy in trigonometric evaluations.
Through angle reduction, the calculations become more manageable, allowing us to direct our efforts towards understanding rather than computation.

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

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15 th of each month are: \(1(9.67), 2(10.72), 3(11.92), 4(13.25)\) \(5(14.37), \quad 6(14.97), \quad 7(14.72), \quad 8(13.77), \quad 9(12.48)\) \(10(11.18), \quad 11(10.00), \quad 12(9.38) . \quad\) The month is represented by \(t,\) with \(t=1\) corresponding to January. A model for the data is $$H(t)=12.13+2.77 \sin \left(\frac{\pi t}{6}-1.60\right)$$. (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

Graph \(f\) and \(g\) in the same coordinate plane. Include two full periods. Make a conjecture about the functions. $$f(x)=\sin x, \quad g(x)=-\cos \left(x+\frac{\pi}{2}\right)$$

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse trigonometric function.

When tuning a piano, a technician strikes a tuning fork for the A above middle \(\mathrm{C}\) and sets up a wave motion that can be approximated by \(y=0.001 \sin 880 \pi t,\) where \(t\) is the time (in seconds). (a) What is the period of the function? (b) The frequency \(f\) is given by \(f=1 / p .\) What is the frequency of the note?

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) \(\mathrm{As} x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) \(\mathrm{As} x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\csc x$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.