/*! 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 35 Use the half-angle formulas to d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 35

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\pi / 8$$

Short Answer

Expert verified
\(\sin(\pi / 8) = \sqrt{\frac{1 - \sqrt{2}/2}{2}}, \cos(\pi / 8) = \sqrt{\frac{1 + \sqrt{2}/2}{2}}, \tan(\pi / 8) = \frac{\sqrt{\frac{1 - \sqrt{2}/2}{2}}}{\sqrt{\frac{1 + \sqrt{2}/2}{2}}}\)

Step by step solution

01

Half-Angle identity for Sine

The half-angle identity for sine is: \[ \sin(\frac{胃}{2}) = 卤\sqrt{\frac{1 - \cos(胃)}{2}}\] We know the value of \(\cos(\pi / 4)\) is \(\sqrt{2}/2\), so we substitute this into the formula to get: \[\sin(\pi / 8) = 卤\sqrt{\frac{1 - \sqrt{2}/2}{2}}\] Since \(\pi / 8\) is in the first quadrant where sine is positive, we take the positive root: \[\sin(\pi / 8) = \sqrt{\frac{1 - \sqrt{2}/2}{2}}\]
02

Half-Angle identity for Cosine

The half-angle identity for cosine is: \[ \cos(\frac{胃}{2}) = 卤\sqrt{\frac{1 + \cos(胃)}{2}}\] Once again, substituting \(\cos(\pi / 4) = \sqrt{2}/2\) gives us: \[\cos(\pi / 8) = 卤\sqrt{\frac{1 + \sqrt{2}/2}{2}}\] As \(\pi / 8\) is in first quadrant where cosine is positive, we take the positive root: \[\cos(\pi / 8) = \sqrt{\frac{1 + \sqrt{2}/2}{2}}\]
03

Tangent of the Angle

The tangent of the angle is given by the ratio of the sine to the cosine: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\] Substituting the previously derived sin and cos values into this formula gives the tangent of the angle: \[ \tan(\pi / 8) = \frac{\sqrt{\frac{1 - \sqrt{2}/2}{2}}}{\sqrt{\frac{1 + \sqrt{2}/2}{2}}}\]

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

Determine whether the statement is true or false. Justify your answer. $$\sin \left(x-\frac{\pi}{2}\right)=-\cos x$$

Verify the identity. $$\cos (n \pi+\theta)=(-1)^{n} \cos \theta, \quad n$ is an integer$$

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

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\sin 5 \theta-\sin 3 \theta$$

Verify the identity. $$\cos \left(\frac{\pi}{3}+x\right)+\cos \left(\frac{\pi}{3}-x\right)=\cos x$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.