Problem 55 - Algebra and Trigonometry

Question

53–58 Find the value of the product or sum. $$\cos 37.5^{\circ} \sin 7.5^{\circ}$$

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{2} - 1}{4} \).

Step 1: Identify the Trigonometric Product-to-Sum Formula

This problem involves finding the value of the product of two trigonometric functions: \( \cos 37.5^{\circ} \sin 7.5^{\circ} \). We can use the product-to-sum formula. Specifically, the formula: \( \cos A \sin B = \frac{1}{2}(\sin(A+B) - \sin(A-B)) \).

Step 2: Substitute Angles into the Formula

Using the formula identified, substitute \( A = 37.5^{\circ} \) and \( B = 7.5^{\circ} \). This gives:\[\cos 37.5^{\circ} \sin 7.5^{\circ} = \frac{1}{2}(\sin(37.5^{\circ} + 7.5^{\circ}) - \sin(37.5^{\circ} - 7.5^{\circ}))\]

Step 3: Simplify the Angles

First calculate \( A+B \) and \( A-B \):- \( A+B = 37.5^{\circ} + 7.5^{\circ} = 45^{\circ} \)- \( A-B = 37.5^{\circ} - 7.5^{\circ} = 30^{\circ} \)Substituting these into the expression yields:\[\frac{1}{2}(\sin 45^{\circ} - \sin 30^{\circ})\]

Step 4: Find Sine of Specific Angles

Use known sine values: \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 5: Substitute Sine Values and Simplify

Substitute the sine values into the expression:\[\frac{1}{2}\left(\frac{\sqrt{2}}{2} - \frac{1}{2}\right)\]Simplifying inside the brackets, we have:\[\frac{\sqrt{2}}{2} - \frac{1}{2} = \frac{\sqrt{2} - 1}{2}\]Thus, the expression becomes:\[\frac{1}{2} \cdot \frac{\sqrt{2} - 1}{2} = \frac{\sqrt{2} - 1}{4}\]

Step 6: Final Answer

Conclude that the value of \( \cos 37.5^{\circ} \sin 7.5^{\circ} \) is \( \frac{\sqrt{2} - 1}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are universally true for all values of the variables involved. They simplify complex trigonometric expressions and are vital in solving trigonometric equations. For instance, in the problem concerning the product of \( \cos 37.5^{\circ} \sin 7.5^{\circ} \), we utilized the product-to-sum formula:- \( \cos A \sin B = \frac{1}{2}(\sin(A+B) - \sin(A-B)) \)

This identity helps convert products of sines and cosines into sums or differences, which often simplifies the calculations involved. Understanding and using these identities effectively can make solving such trigonometric problems much easier.

1. Saves time by simplifying expressions
2. Makes calculations more straightforward

The product-to-sum formula is just one of several identities useful for various mathematical computations and proofs, so it's essential to familiarize yourself with these identities for better mathematical prowess.

Angle Addition and Subtraction

Angle addition and subtraction formulas are crucial tools in trigonometry which are used to derive the sine, cosine, or tangent of a sum or difference of angles. They are particularly helpful when dealing with non-standard angles that don't have simple trigonometric values using conventional tables.In our exercise, we applied these concepts using the formula to change\( \cos 37.5^{\circ} \sin 7.5^{\circ} \) by finding the values of angles \(A+B\) and \(A-B\) as:

• \( A+B = 45^{\circ} \)
• \( A-B = 30^{\circ} \)

Knowing how to quickly compute angle sums and differences allows you to apply the known sine and cosine values effectively. As noted, \( \sin 45^{\circ} \) and \( \sin 30^{\circ} \) can be easily evaluated once the angles are adjusted using these techniques. This makes complex problems simplistic and manageable, even when angles are non-standard or unfamiliar.

Trigonometric Function Values

Understanding the basic values of trigonometric functions at key angles is a foundational aspect of trigonometry. This knowledge allows for the competent use of identities and formulas like the product-to-sum we've used in the given problem.In our steps, knowing the exact values of \( \sin 45^{\circ} \) and \( \sin 30^{\circ} \) was necessary:

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

These values are derived from the properties of special right triangles:- \( 45^{\circ} \) angles stem from an isosceles right triangle, where an equal division of radians gives us the sine and cosine of \( \frac{\sqrt{2}}{2} \).- \( 30^{\circ} \) correspond to the smaller angle of a half-equilateral triangle, offering simple fractional values.These predefined function values turn intricate numerical problems into straightforward arithmetic. Memorize these functional values, as they repeatedly serve as the groundwork for more advanced trigonometric problem-solving.