Problem 182 - Precalculus

Question

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\sin \left(-45^{\circ}\right) \sin \left(-15^{\circ}\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{3} - 1}{4} \).

Step 1: Express Sine Products Using Sum-to-Product Formulas

We use the identity \( \sin A \sin B = \frac{1}{2} [ \cos(A-B) - \cos(A+B) ] \) to express the product \( \sin(-45^{\circ}) \sin(-15^{\circ}) \).

Step 2: Identify Values for A and B

In this problem, \( A = -45^{\circ} \) and \( B = -15^{\circ} \). We will substitute these into the sum-to-product formula.

Step 3: Calculate \( A - B \) and \( A + B \)

Calculate \( A - B = -45^{\circ} - (-15^{\circ}) = -45^{\circ} + 15^{\circ} = -30^{\circ} \) and \( A + B = -45^{\circ} + (-15^{\circ}) = -60^{\circ} \).

Step 4: Substitute and Simplify

Using the formula: \[ \sin(-45^{\circ}) \sin(-15^{\circ}) = \frac{1}{2} [ \cos(-30^{\circ}) - \cos(-60^{\circ}) ] \]

Step 5: Evaluate the Cosines

Find \( \cos(-30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) (since cosine is an even function) and \( \cos(-60^{\circ}) = \cos(60^{\circ}) = \frac{1}{2} \).

Step 6: Calculate the Expression

Substitute the values back into the expression: \[ \frac{1}{2} \left( \frac{\sqrt{3}}{2} - \frac{1}{2} \right) = \frac{1}{2} \left( \frac{\sqrt{3} - 1}{2} \right) = \frac{\sqrt{3} - 1}{4} \]

Step 7: Confirm the Solution

The exact value of \( \sin(-45^{\circ}) \sin(-15^{\circ}) \) is \( \frac{\sqrt{3} - 1}{4} \), and all calculations have been verified.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for solving many problems in trigonometry. They are equations that are true for all values of the involved variables, where each side of the equation represents a trigonometric function. Trigonometric identities include basic ones like the Pythagorean identity, sum and difference formulas, and double angle identities. Among these, the sum-to-product identities are especially useful when transforming the sum or difference of two trigonometric functions into a product. In the exercise at hand, the identity \( \sin A \sin B = \frac{1}{2} [ \cos(A-B) - \cos(A+B) ] \) plays a crucial role in simplifying the product of sines into a more manageable expression.
This identity helps by converting a multiplication of two sines into a subtraction of two cosines, which are often easier to evaluate.

Angle Subtraction

The concept of angle subtraction is vital when dealing with trigonometric identities, particularly those involving the product of sines. It is often necessary to calculate the angles being subtracted or added to apply identities correctly. During angle subtraction:

• Subtract the smaller angle from the larger angle if they are in different quadrants.
• Apply the identity accurately, ensuring correct angle signs are utilized.

In the evaluated exercise, the subtraction \( A - B = -45^{\circ} + 15^{\circ} = -30^{\circ} \) demonstrated the importance of careful angle calculation. Understanding how to switch negative angles to their positive equivalents (or vice versa) simplifies the trigonometric operations since many functions, like cosine, are easier to evaluate in their standardized forms.

Product of Sines

The product of sines involves multiplying two sine values. Often, using basic trigonometric identities can simplify the evaluation of such expressions. For instance, using the sum-to-product identity, the product of sines can be rewritten as a difference or sum of cosines:

• Transforms the multiplication into simpler addition and subtraction operations.
• Helpful for evaluating specific angle problems using known trigonometric values.

In this problem, splitting the product \( \sin(-45^{\circ}) \sin(-15^{\circ}) \) into \( \cos(-30^{\circ}) - \cos(-60^{\circ}) \) by halving the expression helped to easily calculate using well-known values for \( \cos(30^{\circ}) \) and \( \cos(60^{\circ}) \).
This transformation is especially useful since sine and cosine values are more straightforward to find for common angles.

Cosine Function

The cosine function is one of the fundamental components of trigonometry, representing the x-coordinate of the point on the unit circle associated with a given angle. Key characteristics of the cosine function include:

• It is even, meaning \( \cos(-\theta) = \cos(\theta) \).
• Standard values are easy to remember for common angles like 30°, 45°, and 60°.

In this exercise, understanding that \( \cos(-30^{\circ}) \) is equivalent to \( \cos(30^{\circ}) \) helped simplify calculations significantly. The even property of the cosine function eliminates confusion about negative angles, making calculations with angle transformations much more straightforward. Therefore, recognizing these identities can aid in evaluating trigonometric expressions where precise trigonometric values are required.