/*! 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 18 Evaluate (if possible) the sine,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 18

Evaluate (if possible) the sine, cosine, and tangent at the real number. $$t=-\frac{4 \pi}{3}$$

Short Answer

Expert verified
The value of sine at \(t=-\frac{4 \pi}{3}\) is \( \frac{\sqrt{3}}{2}\), the value of cosine is \( -\frac{1}{2}\), and the value of tangent is \( -\sqrt{3}\).

Step by step solution

01

Convert to Positive Angle

As the angle given is negative, convert it to positive by adding \(2\pi\). This is because adding \(2\pi\) or subtracting \(2\pi\) from any angle doesn鈥檛 change its terminal side. Therefore, \(t=-\frac{4 \pi}{3} + 2\pi = \frac{2 \pi}{3}\)
02

Determine Position on the Unit Circle

Locate the angle \( \frac{2 \pi}{3}\) on the unit circle. This angle is in the second quadrant where cosine is negative, and sine is positive.
03

Evaluate Sine, Cosine, and Tangent

Remember that on the unit circle, cosine is the x-coordinate and sine is the y-coordinate of a point. \n\n In this case, the point on the unit circle associated with \( \frac{2 \pi}{3}\) is \(-\frac{1}{2}\), \( \frac{\sqrt{3}}{2}\). Therefore, \n\n \( \sin(t)=\sin(\frac{-4\pi }{3})=\sin(\frac{2\pi }{3})=\frac{\sqrt{3}}{2}\) \n\n \( \cos(t)=\cos(\frac{-4\pi }{3})=\cos(\frac{2\pi }{3})=-\frac{1}{2}\) \n\n The tangent of an angle is the sine divided by the cosine, so \n\n \( \tan(t)=\tan(\frac{-4\pi }{3})=\tan(\frac{2\pi }{3})=\frac{\sin(t)}{\cos(t)}=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=-\sqrt{3}\)

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!

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

True or False? Determine whether the statement is true or false. Justify your answer. The difference between the measures of two coterminal angles is always a multiple of \(360^{\circ}\) when expressed in degrees and is always a multiple of \(2 \pi\) radians when expressed in radians.

Prove that the area of a circular sector of radius \(r\) with central angle \(\theta\) is \(A=\frac{1}{2} \theta r^{2}\) where \(\theta\) is measured in radians.

Prove each identity. (a) \(\arcsin (-x)=-\arcsin x\) (b) \(\arctan (-x)=-\arctan x\) (c) \(\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, \quad x>0\) (d) \(\arcsin x+\arccos x=\frac{\pi}{2}\) (e) \(\arcsin x=\arctan \frac{x}{\sqrt{1-x^{2}}}\)

Use a graphing utility to graph the function. $$f(x)=\arctan (2 x-3)$$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$f(x)=e^{-x} \cos x$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.