/*! 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 86 In Exercises \(61-86,\) use refe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 86

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sin \left(-\frac{35 \pi}{6}\right)$$

Short Answer

Expert verified
The exact value of \(\sin \left(-\frac{35 \pi}{6}\right)\) is \(\frac{1}{2}\).

Step by step solution

01

Convert the Angle to a Value Less than \(2\pi\)

Since \(\sin\) function has a periodicity of \(2\pi\), we can add or subtract multiples of \(2\pi\) to the given angle to get an equivalent angle that lies within one cycle. The given angle is \(-\frac{35 \pi}{6}\). We can convert this to a positive value less than \(2\pi\) by adding \(6\pi\) repeatedly until we get a positive angle less than \(2\pi\). In this case, adding \(6\pi = \frac{36\pi}{6}\) would transform \(-\frac{35 \pi}{6}\) into \(\frac{\pi}{6}\).
02

Find the Reference Angle

The reference angle for an angle \(\theta\) in the first quadrant is simply \(\theta\) itself. Hence, the reference angle for \(\frac{\pi}{6}\) is \(\frac{\pi}{6}\).
03

Compute the Trigonometric Function Value for the Reference Angle

Now, compute the value of sine for the reference angle. Using the unit circle or common trigonometric values, we know that \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).

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!

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

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}+2 x-1 & \text { if } x \geq 2 \\ 3 x+1 & \text { if } x<2\end{array}\right.\) Find \(f(5)-f(-5) . \text { (Section } 1.3, \text { Example } 6)\)

For \(x>0,\) what effect does \(2^{-x}\) in \(y=2^{-x} \sin x\) have on the graph of \(y=\sin x ?\) What kind of behavior can be modeled by a function such as \(y=2^{-x} \sin x ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made an error because the angle I drew in standard position exceeded a straight angle.

The average monthly temperature, \(y,\) in degrees Fahrenheit, for Juneau, Alaska, can be modeled by \(y=16 \sin \left(\frac{\pi}{6} x-\frac{2 \pi}{3}\right)+40,\) where \(x\) is the month of the year \(\quad\) (January \(=1,\) February \(=2, \ldots\) December \(=12\) ). Graph the function for \(1 \leq x \leq 12 .\) What is the highest average monthly temperature? In which month does this occur?

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

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.