Problem 57 - Algebra and Trigonometry

Question

53–58 Find the value of the product or sum. $$\cos 255^{\circ}-\cos 195^{\circ}$$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{\sqrt{2}}{2} \).

Step 1: Convert Angles to Radians

First, we need to convert the given angles from degrees to radians since trigonometric functions often use radians. To convert degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). For \( 255^{\circ} \), it is \( 255 \times \frac{\pi}{180} = \frac{17\pi}{12} \) radians. Similarly, \( 195^{\circ} \) is \( 195 \times \frac{\pi}{180} = \frac{13\pi}{12} \) radians.

Step 2: Apply Cosine Difference Identity

Apply the cosine difference identity: \( \cos(A) - \cos(B) = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \). Here \( A = 255^{\circ} \) and \( B = 195^{\circ} \).

Step 3: Calculate A+B and A-B

Compute \( A + B = 255^{\circ} + 195^{\circ} = 450^{\circ} \) and \( A - B = 255^{\circ} - 195^{\circ} = 60^{\circ} \).

Step 4: Process with the Sine Values

Convert these results to radians. For \( \frac{A+B}{2} = \frac{450}{2} = 225^{\circ} \), convert to radians: \( 225^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{4} \). For \( \frac{A-B}{2} = 30^{\circ} \), convert: \( 30^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6} \).

Step 5: Substitute Values into Identity

For the formula: \( -2 \sin\left(\frac{5\pi}{4}\right) \sin\left(\frac{\pi}{6}\right) \). The exact values are \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

Step 6: Calculate Final Result

Now substitute the sine values into the equation: \( -2 \times \left(-\frac{\sqrt{2}}{2}\right) \times \frac{1}{2} = \frac{\sqrt{2}}{2} \).

Key Concepts

Cosine Difference Identity

The cosine difference identity is a valuable trigonometric tool that helps us simplify the difference between two cosine values of different angles. This identity is specifically written as:

• \( \cos(A) - \cos(B) = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \).

The beauty of this identity lies in its ability to reduce a cosine difference into a product of sine functions. It essentially converts a subtraction into a more manageable multiplication.

This can be particularly useful when the angles are complex and not immediately recognizable. By applying this identity, we can handle trigonometric expressions more comfortably, turning them into simpler problems that require computing known sine values.

In practical applications, this identity is frequently used in mathematical problems involving wave functions, oscillatory motions, and signal processing, where understanding phase differences is crucial.

Angle Conversion

Converting angles from degrees to radians is fundamental in trigonometry, especially since many mathematical applications and software prefer radians. The conversion uses the simple formula:

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Here, using this formula ensures you're correctly interpreting the angle in a way that trigonometric functions expect. For instance, angles like 255° and 195° are converted to radians by multiplying:

• \( 255 \times \frac{\pi}{180} = \frac{17\pi}{12} \),
• \( 195 \times \frac{\pi}{180} = \frac{13\pi}{12} \).

Understanding and performing angle conversion is key for accurately applying trigonometric identities because they work universally in radians. This step can sometimes clear computational hurdles when dealing with standard formulas in scientific calculators or more advanced mathematics.

Moreover, knowing how to convert angles can be handy when interpreting data from real-world scenarios, such as rotating objects or analyzing periodic phenomena.

Sine Values

When applying trigonometric identities, knowing precise sine values for common angles is fundamental. In our exercise, computing accurate sine values for transformed angles is critical for finding the solution.

For example, we first calculated specific intermediary angles, such as:

• \( \frac{A+B}{2} = 225^{\circ} = \frac{5\pi}{4} \)
• \( \frac{A-B}{2} = 30^{\circ} = \frac{\pi}{6} \)

These angles' sine values are:

• \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
• \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)

Using exact values instead of approximations minimizes errors in trigonometric calculations, ensuring precision in your final results.

It's also beneficial to memorize these sine values for commonly encountered angles as they often appear in trigonometry problems. This preparation improves your speed and confidence when tackling more intricate mathematical challenges.