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Chapter 5: Problem 36

Verify each identity. $$\cos (\pi-x)=-\cos x$$

Short Answer

Expert verified
The trigonometric identity \( \cos (\pi-x) = -\cos x \) has been verified successfully.

Step by step solution

01

Substitution of \( \pi - x \) for x in cosine function

In this step, substitute \( \pi - x \) for \(x\) in the cosine function. The result of this substitution transforms the right side of the equation into \( \cos (\pi - x) \).
02

Apply the property of cosine function

The mathematical property of the cosine function shows that \( \cos (\pi - x) = -\cos x \). This means that if \(x\) is replaced with \( \pi - x\) in \( \cos x \), it forms the negative of the original cosine function. Hence, the identity is verified once the necessary substitutions are made and the cosine property applied.

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Key Concepts

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

Cosine Function
The cosine function is a fundamental concept in trigonometry and is often denoted as \( \cos \theta \), where \( \theta \) represents an angle. It is a periodic function that repeats every \( 2\pi \) radians. The basic definition states that cosine of an angle \( \theta \) in the unit circle equals the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
To better understand the cosine function, it's helpful to review its main characteristics:
  • **RANGE and DOMAIN**: Cosine has a range of [-1, 1], and its domain is all real numbers.
  • **PERIODICITY**: It completes one full cycle every \( 2\pi \) radians.
  • **SYMMETRY**: Cosine is an even function, meaning \( \cos(-x) = \cos(x) \).
Understanding these properties makes it easier to apply transformations or substitutions, like \( \pi - x \), when evaluating changes in the cosine value for different angles.
Angle Subtraction
Angle subtraction in trigonometry involves more than just numeric subtraction. It follows specific formulas, particularly when applied to functions like sine and cosine. The angle subtraction formula for cosine is: \[ \cos(a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b \].
For the problem at hand,
  • we substitute \( \pi \) for \( a \) and \( x \) for \( b \), which gives us \( \cos(\pi - x) \).
  • Using known values, we have \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \), simplifying the expression to \( -\cos x \).
This formula allows us to verify identities, such as transforming \( \cos(\pi - x) \) into \( -\cos x \), confirming the manipulations are justified by trigonometric rules.
Trigonometric Properties
Trigonometric properties play a crucial role in solving and verifying identities. These properties are established relationships and rules that trigonometric functions follow. Understanding these can make tasks like verifying identities more intuitive.
Several important properties include:
  • **BASIC IDENTITIES**: Such as \( \sin^2 x + \cos^2 x = 1 \) which relates sine and cosine.
  • **EVEN AND ODD FUNCTIONS**: Cosine is even, while sine is odd, impacting how they respond to negative angles.
  • **PERIODICITY**: As mentioned, cosine's period of \( 2\pi \) means \( \cos(x + 2\pi) = \cos x \).
These properties not only support identity verification, like \( \cos(\pi - x) = -\cos x \), but also aid in simplifying complex expressions and solving equations.

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Most popular questions from this chapter

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \cos 2 x+1=0$$

In Exercises \(160-162,\) solve each equation on the interval \([0,2 \pi)\) Do not use a calculator.\(160.2 \mathrm{cos}\) $$\sin 3 x+\sin x+\cos x=0$$

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$5 \cot ^{2} x-15=0$$

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan x=-5$$

Use words to describe the formula for: the sine of half an angle.
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