/*! 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 27 Find the exact values of the sin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 27

Find the exact values of the sine, cosine, and tangent of the angle. $$-165^{\circ}$$

Short Answer

Expert verified
The exact values for the given angle are: \(\sin(-165^{\circ}) = -\frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(-165^{\circ}) = -\frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(-165^{\circ}) = - (2 - \sqrt{3})\).

Step by step solution

01

Convert Angle to Positive Reference

Since we are given a negative angle, first turn it into a positive equivalent. We do this by adding 360掳 because there are 360掳 in a circle and rotating anticlockwise by 360掳 essentially brings us back to the same position. We get: \(-165^{\circ} +360^{\circ} = 195^{\circ}\)
02

Find the Reference Angle and Identify Quadrant

The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. For \(-165^{\circ}\) or \(195^{\circ}\), the reference angle in the first quadrant is \(180^{\circ} - 195^{\circ} = -15^{\circ}\) or \(360^{\circ} - 195^{\circ} = 165^{\circ}\). We choose the smallest positive angle which is \(15^{\circ}\). The angle is in the third quadrant where sine is negative, cosine is negative, and tangent is positive.
03

Finding the Exact Trigonometric Values

Now we find the trigonometric values of the angle \(15^{\circ}\). It is known that: \(\sin(15^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(15^{\circ}) = 2 - \sqrt{3}\). These values are obtained from half-angle formulas for standard angles.
04

Apply We Use Sine and Cosine Signs for Quadrant III

Finally, we apply the signs of the trigonometric functions for the observed quadrant which is quadrant III. So, the exact values are: \(\sin(-165^{\circ}) = - \sin(15^{\circ}) = -\frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(-165^{\circ}) = - \cos(15^{\circ}) = -\frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(-165^\circ) = \frac{\sin(-165^\circ)}{\cos(-165^\circ)} = -\frac{2 - \sqrt{3}}{1}\).

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

The table shows the average daily high temperatures in Houston \(H\) (in degrees Fahrenheit) for month \(t,\) with \(t=1\) corresponding to January. (Source: National Climatic Data Center) $$ \begin{array}{|c|c|} \hline \text { Month, } t & \text { Houston, } \boldsymbol{H} \\ \hline 1 & 62.3 \\ 2 & 66.5 \\ 3 & 73.3 \\ 4 & 79.1 \\ 5 & 85.5 \\ 6 & 90.7 \\ 7 & 93.6 \\ 8 & 93.5 \\ 9 & 89.3 \\ 10 & 82.0 \\ 11 & 72.0 \\ 12 & 64.6 \\ \hline \end{array} $$ (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above \(86^{\circ} \mathrm{F}\) and below \(86^{\circ} \mathrm{F}\).

Write the expression as the sine, cosine, or tangent of an angle. $$w\sin 3 \cos 1.2-\cos 3 \sin 1.2$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$2 \tan ^{2} x+7 \tan x-15=0$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x-2=\cos x-2$$

Find the exact values of the sine, cosine, and tangent of the angle. $$285^{\circ}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete 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.