Problem 15 - Precalculus Mathematics for Calculus

Question

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 exact value is \( \frac{\sqrt{2}}{2} \).

Step 1: Identify the Formula

The expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) resembles the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

Step 2: Apply the Formula

Substitute \( A = 18^{\circ} \) and \( B = 27^{\circ} \) into the sine addition formula. The expression can be rewritten as \( \sin(18^{\circ} + 27^{\circ}) \).

Step 3: Simplify the Expression

Calculate the angle sum: \( 18^{\circ} + 27^{\circ} = 45^{\circ} \). Thus, the expression becomes \( \sin 45^{\circ} \).

Step 4: Find the Exact Value

The sine of \( 45^{\circ} \) is a known value, which is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Addition Formulas

When working with trigonometric expressions, addition formulas can be incredibly helpful. These formulas allow us to transform and simplify expressions involving the sum of two angles. In this exercise, we recognize that the expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) closely resembles the sine addition formula. This particular addition formula is written as:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

By identifying the pattern in the expression with \( A = 18^{\circ} \) and \( B = 27^{\circ} \), we can rewrite the expression using this formula. By utilizing addition formulas, it's easier to turn a complex looking trigonometric expression into a simple trigonometric function of a single angle.

Subtraction Formulas

Subtraction formulas are another group of useful tools in trigonometry. They allow us to express the difference between two angles in terms of the sine or cosine functions. The general form for sine and cosine subtraction formulas are:

• For sine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
• For cosine: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

Although the exercise we worked on uses an addition formula, it's important to understand subtraction formulas as they are handled in a very similar way. Being comfortable with recognizing when each formula should be applied is key in solving trigonometric problems efficiently. In situations where the expression fits the format of a subtraction formula, substituting the respective values for \(A\) and \(B\) would allow us to simplify the expression meaningfully.

Exact Trigonometric Values

Finding exact trigonometric values is a crucial skill in trigonometry. For common angles such as \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\), these values are well-known and often used in calculations without needing a calculator. In our exercise, after applying the addition formula, we were left with \( \sin 45^{\circ} \). The exact value of \( \sin 45^{\circ} \) is:

• \( \frac{\sqrt{2}}{2} \)

Memorizing these exact values is incredibly useful and saves time. They also allow us to double-check our work by substituting back in known values and verifying our calculations. Understanding and remembering these values enables students to tackle more complex trigonometric problems with confidence.