Problem 69 - Algebra and Trigonometry

Question

\(67-72\). Find the value of the product or sum. $$ \cos 37.5^{\circ} \sin 7.5^{\circ} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{2} - 1}{4} \)

Step 1: Identify the Trigonometric Identity

To find the value of \( \cos 37.5^\circ \sin 7.5^\circ \), we can use the product-to-sum identities. These identities help to express products of sine and cosine functions as sums of trigonometric functions. The identity we will use here is:\[ \cos A \sin B = \frac{1}{2} [ \sin(A+B) - \sin(A-B) ] \]

Step 2: Substitute the Angles into the Identity

In our problem, \( A = 37.5^\circ \) and \( B = 7.5^\circ \). Substitute these values into the identity:\[ \cos 37.5^\circ \sin 7.5^\circ = \frac{1}{2} [ \sin(37.5^\circ + 7.5^\circ) - \sin(37.5^\circ - 7.5^\circ) ] \]

Step 3: Calculate the Angles in the Expression

Calculate the angles in the expression:\[ 37.5^\circ + 7.5^\circ = 45^\circ \]\[ 37.5^\circ - 7.5^\circ = 30^\circ \]

Step 4: Evaluate the Sine Values

Now, find \( \sin 45^\circ \) and \( \sin 30^\circ \):\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \]\[ \sin 30^\circ = \frac{1}{2} \]

Step 5: Substitute Sine Values into the Expression

Substitute the sine values back into the expression:\[ \cos 37.5^\circ \sin 7.5^\circ = \frac{1}{2} [ \frac{\sqrt{2}}{2} - \frac{1}{2} ] \]

Step 6: Simplify the Expression

Simplify the expression inside the brackets:\[ \frac{1}{2} [ \frac{\sqrt{2}}{2} - \frac{1}{2} ] = \frac{1}{2} \cdot \left( \frac{\sqrt{2} - 1}{2} \right) \]\[ = \frac{\sqrt{2} - 1}{4} \]

Step 7: Finalize the Calculation

Finally, we have the expression simplified as:\[ \frac{\sqrt{2} - 1}{4} \]Thus, the value of \( \cos 37.5^\circ \sin 7.5^\circ \) is \( \frac{\sqrt{2} - 1}{4} \).

Key Concepts

Product-to-Sum Identities

Trigonometric identities are formulas that relate the angles and sides of trigonometric functions. When it comes to simplifying products of sine and cosine functions, one valuable set of identities is the product-to-sum identities. These identities transform products into sums or differences, making them easier to handle in calculations.

The specific identity used here is:

• \[ \cos A \sin B = \frac{1}{2} [ \sin(A+B) - \sin(A-B) ] \]

This identity expresses the product of \( \cos A \) and \( \sin B \) as a half of the difference of two sine functions. Applying this transformation helps us break down complex expressions into more manageable parts.

Using product-to-sum identities is particularly useful in integration and simplifying trigonometric equations, allowing us to find explicit values and relations.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They describe the relationship between the angles and the sides of right-angled triangles. Understanding these functions' properties is crucial for tackling various trigonometric problems.

• **Sine Function** \( \sin \theta \)
  • Measures the ratio of the length of the opposite side to the hypotenuse in a right triangle.
  • Values range from -1 to 1.
• **Cosine Function** \( \cos \theta \)
  • Measures the ratio of the length of the adjacent side to the hypotenuse.
  • Values also range from -1 to 1.

These functions are periodic, with a period of \( 360^\circ \) or \( 2\pi \) radians, meaning their values repeat every full circle rotation. This property often allows for predictions on trigonometric patterns and is a fundamental trait used in various applications, from waves to circular motion.

Angle Calculation

Accurately calculating angles is a core part of solving trigonometric problems. In this exercise, we were tasked with finding the value of \( \cos 37.5^\circ \sin 7.5^\circ \) using trigonometric identities.

To apply the product-to-sum identity, angles are added and subtracted:

• \( 37.5^\circ + 7.5^\circ = 45^\circ \)
• \( 37.5^\circ - 7.5^\circ = 30^\circ \)

These calculations are crucial since they directly affect the application of identities. The sine values of these results are then required to finalize the expression:

• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \sin 30^\circ = \frac{1}{2} \)

Knowing common angle values such as these in degrees or radians can significantly ease trigonometric calculations, making it easier to simplify expressions and draw geometric inferences.