Problem 54 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Prove the identity $$\cos (x-\pi)=-\cos x$$ using the angle subtraction formula. Answer: By using the angle subtraction formula and substituting the values for A and B, we can simplify the equation to show that $$\cos (x-\pi)=-\cos x$$, proving the given identity.

Step 1: Write down the angle subtraction formula for cosine

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Step 2: Substitute A = x and B = π

$$\cos (x - \pi) = \cos x \cos\pi + \sin x \sin\pi$$

Step 3: Substitute the values of cos(π) and sin(π)

We know that: - $$\cos \pi = -1$$ - $$\sin \pi = 0$$ Plugging these values into our equation, we get: $$\cos (x - \pi) = \cos x (-1) + \sin x (0)$$

Step 4: Simplify

The equation simplifies as: $$\cos (x-\pi) = -\cos x$$ So, we have successfully proved the given identity.

Key Concepts

Cosine Function

The cosine function is a fundamental component of trigonometry and is used to describe the relationship between the angle and the ratio of the adjacent side to the hypotenuse in a right triangle. It is often denoted as \( \cos \theta \) for an angle \( \theta \) where:

• For a right triangle, \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
• In the unit circle, the cosine of an angle is the x-coordinate of the corresponding point on the circle.

The cosine function is periodic with a period of \( 2\pi \), meaning the function values repeat every \( 2\pi \) radians. This makes the cosine function versatile in various applications, including wave functions and oscillations. In this exercise, understanding the behavior of \( \cos(\pi) \) becomes crucial, as we substitute this in the trigonometric identity. Recognizing that \( \cos(\pi) = -1 \) helps us transform and prove our expression, unraveling the identity given.

Angle Subtraction Formula

The angle subtraction formula is particularly useful in trigonometry when dealing with the cosine of a difference of two angles. The formula is given by:\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]This identity allows us to break down complex trigonometric expressions into simpler components for easy calculation. It is particularly helpful when exact values for trigonometric functions at specific angles are needed.
In this exercise, we used the formula to evaluate \( \cos(x - \pi) \). By substituting \( A = x \) and \( B = \pi \), the formula becomes:

• \( \cos(x - \pi) = \cos x \cos \pi + \sin x \sin \pi \)

With the known values \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \), the transformation was straightforward, leading us to \( \cos(x-\pi) = -\cos x \). This demonstrates the power of trigonometric identities in simplifying and proving expressions.

Mathematical Proofs

Mathematical proofs are a cornerstone of mathematics, providing a rigorous framework to establish the validity of statements using logical reasoning. To effectively craft a proof, certain steps and laws are followed meticulously.

• Identify known facts or identities that are relevant to the problem. For trigonometry, this often involves using well-known identities like the angle subtraction formula.
• Substitute these known values into the equation or identity you are proving.
• Simplify the expression through algebraic manipulation, ensuring each step follows logically from the previous one.

In this exercise, we proved that \( \cos(x-\pi) = -\cos x \) by substituting known values from the angle subtraction formula and simplifying the resulting expression. This step-by-step approach highlights the process of proof, offering clarity and reinforcing understanding. Good proofs are not only correct but also clear and comprehensive, enabling others to follow and understand the reasoning behind the conclusion.