Problem 83 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( 10 \cos 75^\circ \cos 15^\circ \)

Step-by-Step Solution

Answer

The product \( 10 \cos 75^\circ \cos 15^\circ \) can be written as a sum \(2.5\)

Step 1: Identify A and B

In the given problem, \( A = 75^\circ \) and \( B = 15^\circ \).

Step 2: Apply the formula

Use the product-to-sum formula for cosine:\[10 \cos 75^\circ \cos 15^\circ = 10 \cdot \frac{1}{2}[\cos(75^\circ-15^\circ) + \cos(75^\circ+15^\circ)]\]This simplifies to \[5\cos60^\circ + 5\cos90^\circ\]

Step 3: Solve the expression

Use the values of \(\cos60^\circ\) which equals \( \frac{1}{2} \) and \(\cos90^\circ\) which equals 0 to solve the expression:\[5\cos60^\circ + 5\cos90^\circ = 5 \cdot \frac{1}{2} + 5 \cdot 0 = 2.5 + 0 = 2.5\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved and are an essential part of solving various mathematical problems, especially in trigonometry. These identities relate the trigonometric functions—sine, cosine, tangent, and their reciprocals—to one another, allowing complex equations to be simplified and solved more easily.

There are several commonly used trigonometric identities, including the Pythagorean identities, angle sum and difference identities, double angle identities, and product-to-sum formulas. Product-to-sum formulas are particularly useful when converting the product of two trigonometric functions into the sum or difference of two separate functions, which can be easier to evaluate. For example, the exercise provided requires the use of the product-to-sum identity for cosine functions to simplify the expression and find its value.

Cosine Function

The cosine function, one of the primary trigonometric functions, relates the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse. Its value ranges between -1 and 1, and it is often represented as \(\cos\theta\) where \(\theta\) is the angle. Cosine values are particularly important when working with circles and waves and play a vital role in physics and engineering.

In the given exercise, the cosine function appears twice, and its values for certain angles are fundamental constants that can be easily remembered or found on a unit circle. For instance, \(\cos60^\textrm{circ}\) equals \(\frac{1}{2}\), and \(\cos90^\textrm{circ}\) equals 0. Recognizing these values is crucial to solving problems that involve the cosine function, such as the provided textbook exercise, effortlessly.

Angle Sum and Difference Identities

The angle sum and difference identities are pivotal when you need to evaluate the sine or cosine of an angle that is the sum or difference of two known angles. They can simplify complex trigonometric expressions, making it easier to solve for unknown values. These identities state that:

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

Applying these identities, as was done in step 2 of the solution, helps to break down the product of cosines into an addition of cosines of two different angles. By knowing the values of the cosine for these specific angles, the problem can quickly be solved, which is particularly beneficial for students trying to grasp the concept behind trigonometric expressions related to angles.