Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=180^{\circ}, B=45^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(180^{\circ} - 45^{\circ}) \) is \(-\frac{\sqrt{2}}{2}\).

Step 1: Identify the formula

To find \( \cos(A-B) \), we use the formula:\[ \cos(A-B) = \cos A \cdot \cos B + \sin A \cdot \sin B \].

Step 2: Find individual trigonometric values

Evaluate \( \cos 180^{\circ}, \sin 180^{\circ}, \cos 45^{\circ}, \text{and } \sin 45^{\circ} \).- \( \cos 180^{\circ} = -1 \)- \( \sin 180^{\circ} = 0 \)- \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)- \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

Step 3: Substitute the values

Substitute the trigonometric values into the formula:\[ \cos(180^{\circ} - 45^{\circ}) = (-1) \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} \].

Step 4: Simplify the expression

Calculate the simplified result:\[ \cos(180^{\circ} - 45^{\circ}) = -\frac{\sqrt{2}}{2} + 0 = -\frac{\sqrt{2}}{2} \].

Key Concepts

Cosine Formula

The Cosine Formula is a powerful tool in trigonometry used to calculate the cosine of angle differences or sums. For angle differences, the formula is expressed as:

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

It allows us to find the cosine of the difference between two angles, using the cosines and sines of the individual angles.
In simpler terms, it breaks down a complex expression into simpler parts that involve common trigonometric values. This formula is heavily used in scenarios where direct calculation is difficult or impossible.
By applying this formula, we can transform challenging angle calculations into manageable arithmetic problems.

Angle Difference Identity

The Angle Difference Identity is crucial in understanding expressions like \( \cos(180^{\circ} - 45^{\circ}) \).
This identity allows us to express the cosine of a difference as a combination of different trigonometric functions:

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

Here, the product of the cosines adds to the product of the sines. This operation essentially splits the problem into components that are easier to evaluate separately.
Understanding this identity simplifies many problems involving trigonometric expressions, making complex calculations straightforward.

Trigonometric Values

Knowing the exact trigonometric values for common angles is essential in solving these kinds of problems efficiently. Here are the values used in our example:

• \( \cos 180^{\circ} = -1 \)
• \( \sin 180^{\circ} = 0 \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

These values often need to be memorized because they are foundational for many calculations.
While it might seem tedious at first, knowing these directly contributes to how quick and accurate you can solve trigonometric identities.
In the exercise, these values were substituted into the formula to tackle the problem efficiently.

Step-by-Step Problem Solving

Approaching a trigonometric problem in a structured way helps in ensuring accuracy and understanding.
Here's a brief breakdown of the process:1. **Identify the Formula:** Recognize the formula appropriate for the given problem. In this case, it's the cosine angle difference formula.2. **Evaluate Trigonometric Values:** Obtain the exact values for individual angles involved. For example, \( \cos A \) and \( \sin B \).3. **Substitute the Values:** Plug these known values into the identified formula.4. **Simplify the Expression:** Perform calculations to simplify the expression to get the final result.
Following such methodical steps, as shown above, can demystify even the toughest of trigonometry problems. It also provides a clear path to follow, reducing potential errors and confusion.