Problem 101 - Precalculus with Limits

Question

In Exercises 99 - 102, use the sum-to-product formulas to find the exact value of the expression. \( \cos \dfrac{3\pi}{4} - \cos \dfrac{\pi}{4} \)

Step-by-Step Solution

Answer

The exact value of \( \cos \dfrac{3\pi}{4} - \cos \dfrac{\pi}{4} \) is \( \sqrt{2} \)

Step 1: Recall the cosine sum-to-product formula

The cosine sum-to-product formula is defined as \( \cos A - \cos B = -2 \sin \left( \dfrac{A + B}{2} \right) \sin \left( \dfrac{B - A}{2} \right) \)

Step 2: Apply the formula

Substitute \( A = \dfrac{3\pi}{4} \) and \( B = \dfrac{\pi}{4} \) into the formula. The expression becomes \( -2 \sin \left( \dfrac{3\pi/4 + \pi/4}{2} \right) \sin \left( \dfrac{\pi/4 - 3\pi/4}{2} \right) = -2 \sin \dfrac{\pi}{2} \sin -\dfrac{\pi}{4} \)

Step 3: Solve the expression

Remember that \( \sin \dfrac{\pi}{2} = 1 \) and \( \sin -\dfrac{\pi}{4} = -\dfrac{\sqrt{2}}{2} \). Consequently, the expression becomes \( -2 \times 1 \times -\dfrac{\sqrt{2}}{2} = \sqrt{2} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with angles and triangles. These identities are equations involving trigonometric functions that hold true for all values of the involved variables. Learning them helps simplify and solve trigonometric equations. Two common types of trigonometric identities are sum-to-product and product-to-sum formulas. These formulas allow us to rewrite expressions involving sums into products and vice versa. Typically, this simplifies calculations and helps find exact values for more complex expressions.

There are several key trigonometric identities you should know:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Sum and difference formulas for sine and cosine, like \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \).
• Sum-to-product formulas, which include \( \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{B-A}{2} \right) \).

Mastering these identities allows you to manipulate expressions in various ways, helping you to find solutions more efficiently.

Cosine

The cosine function is a fundamental trigonometric function used extensively in mathematics, particularly when dealing with right-angle triangles and unit circles. The basic definition of cosine in right-angle triangles expresses it as the ratio of the length of the adjacent side to the hypotenuse, abbreviated as 'adjacent over hypotenuse.'

In the context of a unit circle, cos(θ) is the x-coordinate of the point on the circle that corresponds to the angle θ measured from the positive x-axis. Because the unit circle has a radius of one, the cosine values range from -1 to 1. Understanding properties of cosine helps in different areas:

• It's periodic with a period of \( 2\pi \).
• It's an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• Simplifies transformations using identities, such as the sum-to-product formula.

Cosine’s role in identities and formulas makes it indispensable in solving trigonometric problems and understanding wave phenomena, geometry, and physics.

Exact Value

In trigonometry, finding the exact value of an expression means determining the precise, non-decimal representation of the value, often using known values of trigonometric functions at specific angles. This is crucial in fields requiring high precision like engineering and computer graphics.

To find exact values, it is helpful to memorize commonly used angles such as \( 0, \pi/6, \pi/4, \pi/3, \pi/2 \) and their sine and cosine values. Knowing these values allows for rapid computation of trigonometric problems without relying on a calculator.

For example, knowing that \( \sin(\pi/2) = 1 \) and \( \sin(-\pi/4) = -\frac{\sqrt{2}}{2} \) aids in evaluating expressions accurately. In the context of the problem given, recognizing these values allowed the calculation of the expression \( \cos \frac{3\pi}{4} - \cos \frac{\pi}{4} \) to yield the exact value \( \sqrt{2} \) seamlessly through substitution and simplified multiplication.