Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \[ \cos(x - \pi) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi \].

Step 2: 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: \[ \cos(x - \pi) = -\cos x \].

Step 3: 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 \( x \).

Key Concepts

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.