Problem 29 - Precalculus: Functions and Graphs

Question

Exer. \(25-36:\) Verify the reduction formula. $$\cos (\theta-\pi)=-\cos \theta$$

Step-by-Step Solution

Answer

The identity \( \cos(\theta - \pi) = -\cos \theta \) is verified.

Step 1: Understand the identity you are given to verify

We need to verify the identity \( \cos (\theta - \pi) = -\cos \theta \). This means that we are trying to prove or show that this identity holds true for any angle \( \theta \).

Step 2: Recall the cosine angle subtraction formula

The cosine of the difference of two angles formula is \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). We will use this formula to expand \( \cos(\theta - \pi) \).

Step 3: Apply the formula to the given expression

Substitute \( A = \theta \) and \( B = \pi \) into the formula: \[\cos(\theta - \pi) = \cos \theta \cos \pi + \sin \theta \sin \pi\] Recall that \( \cos \pi = -1 \) and \( \sin \pi = 0 \).

Step 4: Simplify the expression

Substitute the known values \( \cos \pi = -1 \) and \( \sin \pi = 0 \) into the equation: \[\cos(\theta - \pi) = \cos \theta (-1) + \sin \theta (0)\]This simplifies to \[-\cos \theta + 0 = -\cos \theta\] Hence, the given identity \( \cos(\theta - \pi) = -\cos \theta \) is verified.

Key Concepts

Cosine Subtraction Formula

The cosine subtraction formula is a fundamental trigonometric identity used to find the cosine of the difference between two angles. It is represented as \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). This formula is incredibly useful in trigonometry, especially when dealing with angle transformations and simplifying expressions.

Here's a breakdown of why this formula works:

• It utilizes the properties of cosine and sine functions to express the cosine of an angle difference.
• This identity is derived from the unit circle and the addition formulas, which explain how trigonometric functions interact when angles are added or subtracted.
• The formula is symmetric in the roles of \(A\) and \(B\), making it applicable for any pair of angles.

Remember, this formula will always provide an accurate result when you know the sine and cosine of each individual angle. With this tool in your belt, you can tackle any angle subtraction problem with ease.

Angle Subtraction

Angle subtraction is a technique used in trigonometry to find the trigonometric values of the difference between two specific angles. This technique is crucial because it helps to break down complex trigonometric expressions into simpler forms.

To understand angle subtraction better, consider the expression \( \cos(\theta - \pi) \):

• Here, \(\theta\) is any angle, and \(\pi\) (or 180 degrees) represents a half rotation in the coordinate plane.
• By subtracting \(\pi\) from \(\theta\), you effectively rotate the angle \(\theta\) by 180 degrees to its new position on the unit circle.

Why is this useful? This allows you to work with angles that aren't neatly positioned and to verify trigonometric identities efficiently. It turns the abstract into the understandable, using the geometry of rotation.

Verification of Identities

Verification of identities involves confirming that a proposed equation holds true for all values within its domain, usually through manipulation and application of known formulas. In trigonometry, this process is essential for establishing the validity of new identities or discovering simplifications.

Consider the identity \( \cos(\theta - \pi) = -\cos \theta \). To verify this, we:

• Apply the cosine subtraction formula: \( \cos(\theta - \pi) = \cos \theta \cos \pi + \sin \theta \sin \pi \).
• Use known values: Since \( \cos \pi = -1 \) and \( \sin \pi = 0 \), the expression simplifies directly.
• The result, \( -\cos \theta + 0 = -\cos \theta \), matches the identity we're verifying.

Verifying identities not only strengthens your understanding of trigonometric concepts but also improves problem-solving skills by practicing logical reasoning. It’s like solving a puzzle where every piece must fit just right.