/*! 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 61 Verify the identity. $$\sin \l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 61

Verify the identity. $$\sin \left(\frac{\pi}{2}+x\right)=\cos x$$

Short Answer

Expert verified
The left-hand side of the equation \(\sin \left(\frac{\pi}{2}+x\right)\) does indeed simplify to \(\cos x\). Therefore, \(\sin \left(\frac{\pi}{2}+x\right) = \cos x\) is a true identity.

Step by step solution

01

Understand and use the Addition Formula

The addition formula for sine states that \(\sin(a+b) = \sin a \cos b + \cos a \sin b\). Therefore, we can write \(\sin \left(\frac{\pi}{2}+x\right)\) as \(\sin(\frac{\pi}{2})\cos x + \cos(\frac{\pi}{2})\sin x\).
02

Substitute the values

Substitute the values of \(\sin(\frac{\pi}{2})\) and \(\cos(\frac{\pi}{2})\), which are 1 and 0 respectively. Therefore, the expression becomes \(1 * \cos x + 0 * \sin x\). This simplifies to \(\cos x\).
03

Endpoint Verification

The right-hand side of our initial identity was \(\cos x\). Now the left-hand side, \(\sin \left(\frac{\pi}{2}+x\right)\), has also been simplified down to \(\cos x\) showing that the two sides are indeed equal and that verifies the identity.

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 half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{12}$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=\frac{5}{13}, \quad \pi / 2

Sketch the graph of the function. (Include two full periods.) $$f(x)=\cos (x-\pi)+3$$

Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\sin 6 x+\sin 2 x=0$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{8}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics 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.