/*! 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 66 Use the half-angle formulas to d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 66

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$7 \pi / 12$$

Short Answer

Expert verified
\(\sin(7 \pi / 12) = \sqrt{6} + \sqrt{2}/4\); \(\cos(7 \pi / 12) = \sqrt{6} - \sqrt{2}/4\); \(\tan(7 \pi /12) = \frac{\sqrt{6} + \sqrt{2}/4}{\sqrt{6} - \sqrt{2}/4}\)

Step by step solution

01

Represent \(7 \pi / 12\) as Sum of Known Angles

We begin by expressing \(7 \pi / 12\) as a sum of angles \(\pi / 4\) and \(\pi / 6\) whose trigonometric values we know. So, we have: \(7 \pi / 12 = \pi / 4 + \pi / 6\).
02

Compute Sine

The sum-to-product identity for sine is: \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\). Using this, the sine value is: \[\sin(7 \pi / 12) = \sin(\pi / 4 + \pi / 6) = \sin(\pi / 4)\cos(\pi / 6) + \cos(\pi / 4)\sin(\pi / 6) = \sqrt{2}/2 \cdot \sqrt{3}/2 + \sqrt{2}/2 \cdot 1/2 = \sqrt{6} + \sqrt{2}/4\].
03

Compute Cosine

The sum-to-product identity for cosine is: \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). Using this, the cosine value is: \[\cos(7 \pi / 12) = \cos(\pi / 4 + \pi / 6) = \cos(\pi / 4)\cos(\pi / 6) - \sin(\pi / 4)\sin(\pi / 6) = \sqrt{2}/2 \cdot \sqrt{3}/2 - \sqrt{2}/2 \cdot 1/2 = \sqrt{6} - \sqrt{2}/4\].
04

Compute Tangent

The tangent of an angle is the ratio of its sine to its cosine. Thus, \[\tan(7 \pi /12) = \frac{\sqrt{6} + \sqrt{2}/4}{\sqrt{6} - \sqrt{2}/4}\].

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

Find the exact values of the sine, cosine, and tangent of the angle. $$105^{\circ}=60^{\circ}+45^{\circ}$$

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$12 \sin ^{2} x-13 \sin x+3=0$$

Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Explain what would happen if you divided each side of the equation \(\cot x \cos ^{2} x=2 \cot x\) by \(\cot x .\) Is this a correct method to use when solving equations?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.