Problem 29 - Algebra and Trigonometry

Question

Prove the identity. $$ \cos (x-\pi)=-\cos x $$

Step-by-Step Solution

Answer

The identity \( \cos(x - \pi) = -\cos x \) is proven using the cosine subtraction formula.

Step 1: Understand the Expression

The given problem is to prove the trigonometric identity \( \cos (x-\pi) = -\cos x \). This involves understanding how to manipulate cosine functions and using known trigonometric identities.

Step 2: Apply Cosine Subtraction Formula

We use the formula for cosine of a difference: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Plugging in our values, we have \( \cos(x - \pi) = \cos x \cos \pi + \sin x \sin \pi \).

Step 3: Evaluate Trigonometric Functions

Recall that \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substitute these values into the expression from the previous step: \( \cos(x - \pi) = \cos x (-1) + \sin x (0) \).

Step 4: Simplify the Expression

Simplifying the expression gives us \( \cos(x - \pi) = -\cos x \). Since the sine term vanishes, we are left with just \(-\cos x\).

Step 5: Conclude the Proof

We have manipulated \( \cos(x - \pi) \) to equal \(-\cos x\), thus proving the identity: \( \cos(x - \pi) = -\cos x \).

Key Concepts

Cosine Subtraction Formula

Understanding the cosine subtraction formula is vital for evaluating trigonometric identities like the one in this exercise. The formula, \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), allows us to find the cosine of a difference between two angles. This formula is derived from the properties of the cosine function on a unit circle.

• It helps break down complex angles into more manageable parts by considering each angle separately.
• This approach enables simplification using known cosine and sine values of common angles such as \( \pi \).

The importance of this formula lies in its ability to provide a structured way to approach problems involving the difference of angles. By substituting known values into this formula, we can analyze and simplify the expression to verify identities or solve equations effectively.

Trigonometric Functions Evaluation

Evaluating trigonometric functions, such as finding the value of \( \cos \pi \) and \( \sin \pi \), is fundamental to proving or simplifying expressions. In this exercise, recognizing that \( \cos \pi = -1 \) and \( \sin \pi = 0 \) simplifies the process considerably.

• Tapping into these key values allows for strategic substitutions that make calculations more straightforward.
• Understanding unit circle properties aids in memorizing these crucial angle values.

By knowing these values, we can directly substitute them into the cosine subtraction formula, easing the verification of the identity. This manipulation highlights how prior knowledge of trigonometric function values streamlines problem-solving in identities and equations.

Cosine Function Manipulation

Manipulating cosine functions is crucial in deriving and proving trigonometric identities. In this exercise, the goal is to reach a simplified expression of \( \cos(x - \pi) = -\cos x \). Understanding how to handle and simplify terms gives an edge in tackling such identities.

• Starting with the cosine subtraction formula, substitute known values to break down the expression.
• Simplify by applying basic arithmetic, keeping in mind how multiplication with zero or negative values affects the result.

This process showcases the logical steps needed to manipulate trigonometric functions accurately. By systematically working through the expression, the final simplified form confirms the identity, showing mastery of cosine function manipulation.