/*! 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 16 Find the exact values of the sin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 16

Find the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}=135^{\circ}+30^{\circ}$$

Short Answer

Expert verified
\(\sin{165^{\circ}} = \frac{-1 + \sqrt{3}}{2\sqrt{2}}\), \(\cos{165^{\circ}} = -\frac{1 + \sqrt{3}}{2\sqrt{2}}\), and \(\tan{165^{\circ}} = -2 - \sqrt{3}\)

Step by step solution

01

Application of Trigonometric Identities

Write down the trigonometric function identities for sine, cosine and tangent of sum of two angles. The identities are: \(\sin(a + b) = \sin{a}\cos{b} + \cos{a}\sin{b}\), \(\cos(a + b) = \cos{a}\cos{b} - \sin{a}\sin{b}\) and \(\tan(a + b) = \frac{\tan{a} + \tan{b}}{1 - \tan{a}\tan{b}}\).
02

Substitution

Substitute the angles into the identities. Here, \(a = 135^{\circ}\) and \(b = 30^{\circ}\). Using the identities, calculate the exact values of sine, cosine and tangent of 165 degrees.
03

Computation

Compute the values using the known exact values of sine, cosine, and tangent of 30 degrees and 135 degrees. The known exact values are: \(\sin{30^{\circ}} = 0.5\), \(\cos{30^{\circ}} = \frac{\sqrt{3}}{2}\), \(\tan{30^{\circ}} = \frac{1}{\sqrt{3}}\), \(\sin{135^{\circ}} = \frac{\sqrt{2}}{2}\), \(\cos{135^{\circ}}=-\frac{\sqrt{2}}{2}\), and \(\tan{135^{\circ}} = -1\). Now substitute these values into the identities and compute.

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

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$

Use the sum-to-product formulas to find the exact value of the expression. $$\cos 120^{\circ}+\cos 60^{\circ}$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\tan \frac{x}{2}-\sin x=0$$

Determine whether the statement is true or false. Justify your answer. \(\sin \frac{u}{2}=-\sqrt{\frac{1-\cos u}{2}}\) when \(u\) is in the second quadrant.

Proof (a) Write a proof of the formula for \(\sin (u+v)\) (b) Write a proof of the formula for \(\sin (u-v)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Decision 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.