Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=270^{\circ}, B=45^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \(\cos(270^{\circ} + 45^{\circ})\) is \(\frac{\sqrt{2}}{2}\).

Step 1: Recall Cosine Addition Formula

The formula for the cosine of the sum of two angles, \(\cos(A+B)\), is given by: \[\cos(A+B) = \cos A \cos B - \sin A \sin B.\]

Step 2: Determine Trigonometric Values for 270° and 45°

Find the cosine and sine for each angle: - For \(A = 270°\), \(\cos 270° = 0\) and \(\sin 270° = -1\).- For \(B = 45°\), \(\cos 45° = \frac{\sqrt{2}}{2}\) and \(\sin 45° = \frac{\sqrt{2}}{2}\).

Step 3: Substitute Values into the Formula

Substitute the trigonometric values into \(\cos(A+B) = \cos A \cos B - \sin A \sin B\): \[\cos(270° + 45°) = (0)(\frac{\sqrt{2}}{2}) - (-1)(\frac{\sqrt{2}}{2}).\]

Step 4: Simplify the Expression

Calculate the expression: - \(0 \times \frac{\sqrt{2}}{2} = 0\) - \(-1 \times \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}\). Since it's subtraction, \(0 + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}\).

Step 5: Conclusion: Final Result

The exact value of \(\cos(A+B)\) for \(A = 270^{\circ}\) and \(B = 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the trigonometric functions that are true for all angles. These identities are crucial tools in simplifying expressions and solving equations. One of the most useful identities in trigonometry is the cosine addition formula:

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

This particular identity allows us to find the cosine of the sum of two angles. It breaks down the problem into parts that are easier to manage by using the individual cosine and sine values. It's important to note that trigonometric identities help in deriving other identities and formulas, as they form the backbone of trigonometric transformations.
Learning and memorizing these identities can be very beneficial, especially when dealing with complex trigonometric problems.

Exact Trigonometric Values

Trigonometry often requires finding exact values for sine, cosine, and tangent of important angles such as 0°, 30°, 45°, 60°, and 90°. These values are easy to remember because they repeat regularly and are often used across various problems.

• For example, \(\cos 45° = \sin 45° = \frac{\sqrt{2}}{2}\), making it a central value for calculations.
• For angles like \(270°\), it's essential to recognize \(\cos 270° = 0\) and \(\sin 270° = -1\). This is because they represent significant points on the unit circle's axes.

Memorizing these values is crucial. They make solving problems involving trigonometric identities more straightforward because you can substitute these known values for quick, exact calculations.
Moreover, understanding these values aids in quickly obtaining results and verifying the correctness of answers you derive using identities or other trigonometric methods.

Angle Addition in Trigonometry

The concept of angle addition in trigonometry revolves around calculating the sine, cosine, or tangent of the sum or difference of two angles. This is particularly helpful when dealing with non-standard angles, allowing us to decompose them into known quantities. Here's how angle addition formulas, like the cosine addition formula, benefit solving problems:

• They break the sum of two angles into components that are easier to calculate using known values.
• For example, knowing \(\cos(270° + 45°)\) can be calculated by using \(\cos(270°)\) and \(\cos(45°)\), along with their sine counterparts.

By thoroughly understanding angle addition and its formulas, you can handle more complex problems and become equipped to tackle angles beyond those typically memorized, by uncovering new relationships.
The approach of using triangles or the unit circle to visualize and derive these values makes this concept not only more accessible but also intuitive once you've practiced it enough.