Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the product as a sum or difference. \(\frac{5}{2} \sin \frac{3 \pi}{4} \sin \frac{5 \pi}{6}\)

Step-by-Step Solution

Answer

The given product can be written as a sum or difference of cosines, and the simplified form is \(\frac{5}{4} [cos(\frac{\pi}{12}) - cos(\frac{11 \pi}{12})]\).

Step 1: Identify the values

First identify what A and B are by looking at the given problem. Here, \(A = \frac{3 \pi}{4}\) and \(B = \frac{5 \pi}{6}\).

Step 2: Apply the Product-to-Sum formula

Now apply the Product-to-Sum formula for sin A sin B. Substitute A and B into the formula: \(\frac{5}{2} \cdot \frac{1}{2} [cos(\frac{3 \pi}{4} - \frac{5 \pi}{6}) - cos(\frac{3 \pi}{4} + \frac{5 \pi}{6})]\). The constant multipliers outside the expression are to follow the rule as well, and it will be left outside, giving: \(\frac{5}{4} [cos(\frac{3 \pi}{4} - \frac{5 \pi}{6}) - cos(\frac{3 \pi}{4} + \frac{5 \pi}{6})]\).

Step 3: Simplify the expression

Simplify the subtracted and added radian measures. This gives: \(\frac{5}{4} [cos(-\frac{\pi}{12}) - cos(\frac{11 \pi}{12})] \). Note: The negative angle can be turned into positive with cosine since cosine has even symmetry.

Step 4: Simplify further

Now simplify the expression further. The cosine of negative angle will turn positive. So, the expression is simplified to: \(\frac{5}{4} [cos(\frac{\pi}{12}) - cos(\frac{11 \pi}{12})]\)

Key Concepts

Trigonometric identities

Trigonometric identities are equations that are true for all possible values of the trigonometric function variables. They are used to simplify expressions, solve trigonometry problems, and prove other mathematical theorems. Examples of basic trigonometric identities include the Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and the even-odd identities, which state that \( \sin(-\theta) = -\sin(\theta) \) and \( \cos(-\theta) = \cos(\theta) \).

In our example, knowledge of trigonometric identities helps us to understand that cosine is an even function—meaning \( \cos(-\theta) = \cos(\theta) \)—allowing us to simplify the expression in the final step of the provided solution.

Sum and difference trigonometric formulas

The sum and difference formulas are powerful tools in trigonometry that allow us to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. They are essential for simplifying complex trigonometric expressions.

For our exercise, we specifically utilize the product-to-sum identities, which are derived from the sum and difference formulas. The product-to-sum formula for sine that we used is: \( \sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \).

This formula transforms the product of sines into a sum and difference of cosines, greatly simplifying the expression and hence making it easier to evaluate.

Trigonometry problem solving

Solving trigonometry problems often involves a systematic approach that includes identifying applicable formulas, substituting correctly, and then simplifying. The first step in the provided exercise was to identify values of A and B from the expression.

Once these are known, the relevant product-to-sum formula is applied. The careful execution of this step ensures that the constants are treated correctly, as seen with the multiplier \( \frac{5}{2} \), which is factored outside of the trigonometric functions. Finally, the problem is simplified by using trigonometric identities and arithmetic skills to arrive at a solution that is much easier to comprehend and compute.

Sine and cosine functions

The sine and cosine functions are fundamental in trigonometry and are defined for all real numbers. The sine function \( \sin(\theta) \) represents the y-coordinate of a point on the unit circle at a given angle \( \theta \) from the positive x-axis, while the cosine function \( \cos(\theta) \) represents the x-coordinate.

These two functions have specific symmetries and periodic behaviors that are useful in simplifying expressions. For instance, cosine's even symmetry is employed in our exercise to turn \( \cos(-\frac{\pi}{12}) \) into \( \cos(\frac{\pi}{12}) \) without changing the value of the expression. Understanding how these functions behave and interact is crucial in the study of trigonometry and in solving problems such as the one presented.