Problem 53 - Algebra and Trigonometry

Question

53–58 Find the value of the product or sum. $$2 \sin 52.5^{\circ} \sin 97.5^{\circ}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt{2} + \sqrt{3}}{2} \)

Step 1: Recognize the Trigonometric Identity

We recognize that the expression involves the product of two sine functions. We can use the product-to-sum identity: \( 2 \sin A \sin B = \cos (A-B) - \cos (A+B) \) to transform this product into a sum.

Step 2: Apply the Identity

Substitute \( A = 52.5^{\circ} \) and \( B = 97.5^{\circ} \) into the identity: \[ 2 \sin 52.5^{\circ} \sin 97.5^{\circ} = \cos(52.5^{\circ} - 97.5^{\circ}) - \cos(52.5^{\circ} + 97.5^{\circ}) \].

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

Compute \( A - B \) and \( A + B \):- \( 52.5^{\circ} - 97.5^{\circ} = -45^{\circ}\)- \(52.5^{\circ} + 97.5^{\circ} = 150^{\circ}\)

Step 4: Substitute Calculated Angles

Substitute the calculated angles into the expression: \( \cos (-45^{\circ}) - \cos (150^{\circ}) \).

Step 5: Evaluate the Cosine Functions

Use the properties of cosine functions:- \( \cos(-45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)- \( \cos(150^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2} \)

Step 6: Compute the Final Expression

Substitute the calculated values to find the result: \[ \frac{\sqrt{2}}{2} - \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{2} + \sqrt{3}}{2} \].

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate various trigonometric functions to one another. These identities can simplify expressions and solve trigonometric equations efficiently. One commonly used set of identities is the product-to-sum formulas. These formulas transform the product of sines and cosines into sums or differences, making calculations simpler.

• The product-to-sum formula for sine is: \( 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \). This is particularly useful when needing to simplify the expression involving a product of two sine functions.

Understanding these identities can greatly facilitate solving complex trigonometric problems by reducing them into simpler forms that are easier to interpret and evaluate.

Sine Function

The sine function is one of the fundamental functions in trigonometry. It describes the ratio of the length of the side opposite an angle to the hypotenuse in a right-angled triangle.

• It is denoted as \( \sin \theta \) where \( \theta \) is the angle.
• Sine is a periodic function with a period of \( 360^{\circ} \, (2\pi \, ext{radians}) \), which means it repeats its values every full rotation around the unit circle.

A unique feature of the sine function is its symmetry:

• The function is odd, meaning \( \sin(-\theta) = -\sin(\theta) \).
• It has a range from \(-1\) to \(1\).

These properties help in evaluating sine at different angles, crucially aiding in solving problems involving angles and their transformations.

Cosine Function

The cosine function, like the sine function, is a fundamental part of trigonometry. It relates to the cosine of an angle, providing a ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• It is usually denoted as \( \cos \theta \).
• Cosine is also periodic with a period of \( 360^{\circ} \, (2\pi \, ext{radians}) \).

Notably, the cosine function is even, which means it satisfies the property \( \cos(-\theta) = \cos(\theta) \). Its range, similar to sine, is between \(-1\) and \(1\).

• Important cosine angles include \( \cos 0^{\circ} = 1 \) and \( \cos 90^{\circ} = 0 \).

These characteristics, especially its even nature, are vital when working with expressions like \( \cos(150^{\circ}) \) or even negative angles.

Angle Calculation

Calculating angles plays a critical role in applying trigonometric identities. When dealing with functions of angles, it's necessary to precisely compute resultant angles involved in identities like the product-to-sum.Steps to calculate new angles:

• Subtraction and addition of angles is fundamental, as shown when calculating \( A - B \) and \( A + B \) using given \( A = 52.5^{\circ} \) and \( B = 97.5^{\circ} \).
• Handling negative angles is also important. Remember, \( \cos \) of a negative angle, \( \cos(-45^{\circ}) \), equals \( \cos(45^{\circ}) \), due to its property as an even function.

Performing accurate angle calculations ensures the integrity of subsequent trigonometric evaluations and keeps the problem-solving approach precise.