Problem 71 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Trigonometric Identity

To solve the expression \( \cos 255^\circ - \cos 195^\circ \), we notice that it can be simplified using a trigonometric identity. The relevant identity here is the sum-to-product identity for cosines: \( \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \).

Step 2: Substitute the Given Angles into the Identity

Substitute \( A = 255^\circ \) and \( B = 195^\circ \) into the identity. Calculate \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \). \[ \frac{255^\circ + 195^\circ}{2} = 225^\circ \]\[ \frac{255^\circ - 195^\circ}{2} = 30^\circ \]

Step 3: Calculate Each Sine Component

Calculate \( \sin(225^\circ) \) and \( \sin(30^\circ) \).- \( \sin(225^\circ) = -\frac{\sqrt{2}}{2} \) since 225 degrees is in the third quadrant.- \( \sin(30^\circ) = \frac{1}{2} \).

Step 4: Substitute Back into the Identity and Solve

Substitute the calculated sine values back into the identity:\[-2 \sin \left( 225^\circ \right) \sin \left( 30^\circ \right) = -2 \left( -\frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right)\]Simplify the expression:\[= 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \]\[= \frac{\sqrt{2}}{2}\]

Step 5: State the Final Result

Thus, the value of the expression \( \cos 255^\circ - \cos 195^\circ \) is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Sum-to-Product Identity

Trigonometric identities are essential tools in simplifying and solving trigonometric equations. One powerful identity is the sum-to-product identity. It allows you to express sums or differences of trigonometric functions as products. For cosines, the identity is:

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

This can be quite helpful when you have an expression involving the difference of two cosines and need to simplify it into a product form. By transforming the expression, calculations often become more straightforward, and this can reveal hidden simplifications.
In the case of the expression \( \cos 255^\circ - \cos 195^\circ \), this identity is employed to transform the difference into a product of sines, making the final evaluation more manageable.

Cosine Differences

Differences of cosines like \( \cos 255^\circ - \cos 195^\circ \) can look daunting at first. However, with the right trigonometric identity, they become easier to evaluate. By recognizing that the difference of two cosine values can be perceived as part of a pattern consistent with the sum-to-product identities, the problem simplifies itself down.
When using the identity:

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

we see that you do not directly calculate the cosines, instead, you transform it into the sines of half the sum and half the difference of the angles. This reformation can make more difficult problems approachable and solvable through recognitions and substitutions.

Angle Reduction

Angle reduction is a technique used to make trigonometric calculations simpler. Often, it involves transforming an angle that is not immediately easy to work with into one that is familiar or has simpler sine or cosine values. For example, in the given solution:

• We calculated \( \frac{255^\circ + 195^\circ}{2} = 225^\circ \) and determined the sine of this angle.
• Angles like \( 225^\circ \), which are typical on the unit circle, have known values such as \( \sin(225^\circ) = -\frac{\sqrt{2}}{2} \).

Similarly, the calculation of \( \frac{255^\circ - 195^\circ}{2} = 30^\circ \) reduces the angle to a familiar value with known results, \( \sin(30^\circ) = \frac{1}{2} \), thus reducing the complexity and easing the calculation process.
By utilizing angle reduction, one can transform a potentially unfriendly angle computation into a straightforward one, making your work much more efficient and clear-cut.