Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=270^{\circ}, B=30^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(270^{\circ}+30^{\circ}) \) is \( \frac{1}{2} \).

Step 1: Understand the Formula

To find the value of \( \cos(A+B) \), we use the cosine addition formula: \( \cos(A+B) = \cos A \cos B - \sin A \sin B \). This formula helps calculate the cosine of the sum of two angles, \( A \) and \( B \).

Step 2: Substitute Angle Values

Substitute the given angles into the formula: \( \cos(270^{\circ} + 30^{\circ}) = \cos 270^{\circ} \cos 30^{\circ} - \sin 270^{\circ} \sin 30^{\circ} \).

Step 3: Find Trigonometric Functions

Determine the trigonometric values: \( \cos 270^{\circ} = 0 \), \( \sin 270^{\circ} = -1 \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 4: Substitute Values Into Formula

Substitute the trigonometric values into the cosine addition formula: \( 0 \cdot \frac{\sqrt{3}}{2} - (-1) \cdot \frac{1}{2} \).

Step 5: Simplify the Expression

Compute the expression: \( 0 - (-\frac{1}{2}) = 0 + \frac{1}{2} = \frac{1}{2} \). This gives the exact value of \( \cos(270^{\circ} + 30^{\circ}) = \frac{1}{2} \).

Key Concepts

Cosine Addition Formula

The Cosine Addition Formula is a vital tool in trigonometry that helps to determine the cosine of the sum of two angles. This powerful formula is given as:

• \( \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \)

To use this formula effectively, you need to know the individual cosine and sine values of angles \(A\) and \(B\). Once these values are found, they are plugged into the formula to calculate \(\cos(A + B)\). This method is not only applicable in theoretical problems but also has practical uses in physics and engineering. Calculating \(\cos(A + B)\) becomes straightforward once you have the exact trigonometric values.

Exact Trigonometric Values

Trigonometry relies heavily on specific angle values that frequently appear in calculations. These are called exact trigonometric values, and they simplify the process of solving problems involving trigonometric functions.

• Some key angles include \(0^\circ, 30^\circ, 45^\circ, 60^\circ, \) and \(90^\circ\).
• For example, \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\).

For more complex problems, it is important to memorize these values or have a way to derive them quickly. In the problem above, knowing that \(\cos 270^\circ = 0\) and \(\sin 270^\circ = -1\) allowed for a fast substitution into the Cosine Addition Formula. Having these values ready can greatly enhance your efficiency in tackling trigonometric calculations.

Angle Addition in Trigonometry

Angle addition is a concept in trigonometry that involves summing angles to find the related trigonometric function's value of the result. The most common functions used include sine and cosine. In practice, angle addition is helpful to solve complex trigonometric equations, analyze periodic functions, and even apply in fields such as physics. To work with angle addition:

• Identify if you're working with \(\sin(A + B)\) or \(\cos(A + B)\).
• Use the respective addition formula: for cosine, apply \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).

Angles like 270 degrees and 30 degrees are broken down to find each component's sine and cosine, which is essential when applying these formulas. Mastering angle addition and its formulas can simplify many trigonometric calculations significantly, allowing you to focus on more complex problems.