Problem 13 - Algebra and Trigonometry

Question

\(13-18\) m Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. \(\sin 18^{\circ} \cos 27^{\circ}+\cos 18^{\circ} \sin 27^{\circ}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 1: Identify the Trigonometric Identity

Recognize that the expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) can be rewritten using the sine addition formula: \( \sin(a+b) = \sin a \cos b + \cos a \sin b \).

Step 2: Set the Values for the Identity

Compare the expression to the identity \( \sin(a + b) \) and identify \( a = 18^{\circ} \) and \( b = 27^{\circ} \). This means we can rewrite the expression as \( \sin(18^{\circ} + 27^{\circ}) \).

Step 3: Simplify the Angle Sum

Calculate the sum inside the sine function: \( 18^{\circ} + 27^{\circ} = 45^{\circ} \). Therefore, the expression simplifies to \( \sin 45^{\circ} \).

Step 4: Find the Exact Value of the Trigonometric Function

Recall the exact value for \( \sin 45^{\circ} \). From trigonometric values of special angles, \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Key Concepts

Sine Addition Formula

The sine addition formula is an essential tool in trigonometry that helps simplify expressions involving trigonometric functions. Specifically, this identity takes two angles and forms a single sine function: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]In our exercise, we used this to combine the angles 18° and 27°. By recognizing that \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) matches the pattern of the sine addition formula, we can simplify it to \( \sin(18^{\circ} + 27^{\circ}) \).

• This formula is invaluable for restructuring multiple angle terms into a single trigonometry function.
• It helps in calculations of angles that are not typically found on a standard trigonometric table.

Understanding this identity empowers you to tackle a wide variety of trigonometric problems confidently.

Exact Value of Trigonometric Functions

In trigonometry, knowing the exact values for specific angles is crucial for calculations. Instead of approximations, these `exact values` provide precise answers, often in terms of radicals or simple fractions. When the expression simplified to \( \sin 45^{\circ} \), a special angle, we referred to the known exact value: \[ \sin 45^{\circ} = \frac{\sqrt{2}}{2} \]Reasons to prefer exact values include:

• High precision in mathematical computations.
• Less rounding error, which is crucial in fields like physics and engineering.
• Solutions remain consistent and reproducible across different contexts.

By memorizing a few key exact values, you can easily solve many trigonometric problems without a calculator.

Special Angles in Trigonometry

Special angles like 30°, 45°, and 60° are pivotal in trigonometry. They form the backbone of many fundamental identities and exact values. Understanding and memorizing their trigonometric values allow for quick evaluations and simplify complex expressions. For example:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• Other angles like 0°, 90°, 180°, and 360° also offer straightforward trigonometric values.

These angles relate closely to geometric constructs like the unit circle, an essential concept connecting angles directly to their trigonometric values.Focusing on these special angles and their trigonometric relationships enables a deeper understanding of trigonometry and enhances problem-solving efficiency.