/*! 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 34 Show that \(\cos (x-\pi)=-\cos x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 34

Show that \(\cos (x-\pi)=-\cos x\) for all \(x\).

Short Answer

Expert verified
\(\cos(x-\pi)=-\cos x\) for all \(x\) by applying the cosine difference formula and trigonometric values.

Step by step solution

01

Use the Cosine Difference Formula

To find \( \cos(x - \pi) \), we use the cosine difference formula, which states: \[ \cos(a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b \]. In this exercise, let \( a = x \) and \( b = \pi \). Thus, the expression becomes: \[ \cos(x - \pi) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi \].
02

Apply Trigonometric Values for \(\pi\)

We know from trigonometric values that \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substitute these values into the expression from Step 1: \[ \cos(x - \pi) = \cos x \cdot (-1) + \sin x \cdot 0 \]. This simplifies to: \[ \cos(x - \pi) = -\cos x \].
03

Confirm the Expression

After substituting the trigonometric values and simplifying, we find that \( \cos(x - \pi) = -\cos x \). Therefore, we have shown that the given expression is true for all \( x \).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Difference Formula
The Cosine Difference Formula is a powerful tool used in trigonometry to find the cosine of the difference between two angles. It tells us that for any angles \(a\) and \(b\), the cosine of their difference is given by:
  • \(\cos(a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b\).
This formula is particularly useful when we want to express the cosine of an angle in terms of trigonometric functions of other angles. In the given exercise, this formula transforms \(\cos(x - \pi)\) into a combination of \(\cos x\) and \(\sin x\), making it easier to simplify by using known trigonometric values.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the involved variables. They are the foundation of solving a wide range of trigonometry problems and helping derive new formulas. Some basic identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities.In this exercise, we used the cosine and sine functions specifically for \(\pi\) to simplify the expression \(\cos(x - \pi)\). The identities help to make complex trigonometric expressions more manageable and to verify the truth of equations like the one in our task.
Trigonometric Values
Trigonometric functions have specific values at well-known angles. For the angle \(\pi\), these values are important. We know that:
  • \(\cos \pi = -1\)
  • \(\sin \pi = 0\)
Using these trigonometric values helps simplify expressions involving \(\pi\). In our step-by-step solution, substituting these values allowed us to transform \(\cos(x - \pi) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi\) into \(\cos(x - \pi) = -\cos x\), showing how easily known values help in simplifying complex trigonometric expressions.
Proof in Trigonometry
Proof in trigonometry involves demonstrating that certain trigonometric relationships or equations are true for all applicable values. It requires a solid understanding of trigonometric formulas, identities, and values.In our problem, proving \(\cos(x - \pi) = -\cos x\) involves applying the cosine difference formula, substituting specific trigonometric values, and simplifying the equation. This demonstrates a clear and logical step-by-step method to establish the truth of the given expression for any angle \(x\). Each step builds the case, confirming the integrity of the trigonometric proof.

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 the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=\tan \left(2 x-\frac{\pi}{3}\right) $$

Convert the following radian measures to degrees. (a) \(\frac{7}{6} \pi\) (b) \(\frac{3}{4} \pi\) (c) \(-\frac{1}{3} \pi\) (d) \(\frac{4}{3} \pi\) (e) \(-\frac{35}{18} \pi\) (f) \(\frac{3}{18} \pi\)

Find the value of \(k\) such that the line \(k x-3 y=10\) (a) is parallel to the line \(y=2 x+4\); (b) is perpendicular to the line \(y=2 x+4\); (c) is perpendicular to the line \(2 x+3 y=6\).

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=3 \cos \left(x-\frac{\pi}{2}\right)-1 $$

Calculate. (a) \(\frac{56.3 \tan 34.2^{\circ}}{\sin 56.1^{\circ}}\) (b) \(\left(\frac{\sin 35^{\circ}}{\sin 26^{\circ}+\cos 26^{\circ}}\right)^{3}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.