/*! 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 51 Use the half-angle formulas to d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 51

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Short Answer

Expert verified
The exact values based on the half-angle formulas are \[sin(75^{\circ})=\frac{\sqrt{4 - 2\sqrt{3}}}{2}\], \[cos(75^{\circ})=\frac{\sqrt{4 + 2\sqrt{3}}}{2}\] and \[tan(75^{\circ})= \sqrt{1 - \sqrt{3}}\]

Step by step solution

01

Calculate the Cosine

First, express the angle of 75 degrees as the sum of two known angles for which we can easily find the cosine. One combination is 45 and 30 degrees. Using the sum of angles identity for cosine, \[cos(x + y) = cos(x)cos(y) - sin(x)sin(y)\], we can calculate: \[cos(75) = cos(45 + 30) = cos(45)cos(30) - sin(45)sin(30) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\]
02

Apply Half-Angle Formulas for Sine and Cosine

Next, we substitute the found cosine value into the half-angle formulas for sine and cosine: \[sin\left(\frac{75}{2}\right) = \sqrt{\frac{1 - cos(75)}{2}} = \sqrt{\frac{1 - (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} = \sqrt{\frac{2 - \sqrt{6} - sqrt{2}}{4}} = \frac{\sqrt{4 - 2\sqrt{3}}}{2}\], \[cos\left(\frac{75}{2}\right) = \sqrt{\frac{1 + cos(75)}{2}} = \sqrt{\frac{1 + (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} = \sqrt{\frac{2 + \sqrt{6} + sqrt{2}}{4}} = \frac{\sqrt{4 + 2\sqrt{3}}}{2}\]
03

Apply Half-Angle Formula for Tangent

Lastly, substitute the found cosine value into the half-angle formula for the tangent: \[tan\left(\frac{75}{2}\right) = \sqrt{\frac{1 - cos(75)}{1 + cos(75)}} = \sqrt{\frac{1 - (\frac{\sqrt{6} + sqrt{2}}{4})}{1 + (\frac{\sqrt{6} + \sqrt{2}}{4})}} = \sqrt{1 - \sqrt{3}}\]

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

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\tan \frac{u}{2}=\csc u-\cot u$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$

Write the trigonometric expression as an algebraic expression. $$\sin (2 \arcsin x)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin 3 y}{\sin y}=1-2 \sin ^{2} y+2 \cos ^{2} y$$

Find the solutions of the equation in the interval \([0,2 \pi)\). Use a graphing utility to verify your answers. $$\sin \frac{x}{2}+\cos x-1=0$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and 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.