Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$4 \sin 15^{\circ} \cos 15^{\circ}$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Identify the applicable trigonometric identity

Recognize that the expression \(4 \sin 15^{\circ} \cos 15^{\circ}\) resembles the form of a double angle formula. The double angle identity for sine is \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \).

Step 2: Use the double angle identity

Since the double angle identity for sine is \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we can see that \( 2 \sin(15^{\circ}) \cos(15^{\circ}) = \sin(30^{\circ}) \). So \( 4 \sin(15^{\circ}) \cos(15^{\circ}) = 2 \times 2 \sin(15^{\circ}) \cos(15^{\circ}) = 2 \sin(30^{\circ}) \).

Step 3: Evaluate the sine function

We know that \( \sin(30^{\circ}) = \frac{1}{2} \). Thus, we can substitute this into our expression from Step 2. Therefore, \( 2 \sin(30^{\circ}) = 2 \times \frac{1}{2} = 1 \).

Step 4: Write the final answer as a single number

Conclude that \( 4 \sin 15^{\circ} \cos 15^{\circ}\) simplifies to \( 1 \) using the trigonometric identity for sine of double angle.

Key Concepts

Double Angle Formula

The double angle formula is a useful identity in trigonometry that simplifies expressions where angles are doubled. It's especially helpful when dealing with trigonometric functions such as sine, cosine, and tangent. In our example, we focused on the double angle formula for sine:

• \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \)

This identity tells us how we can express the sine of a double angle in terms of the sine and cosine of the original angle \( \theta \). It comes in handy when you need to transform a complex expression into something simpler or when you need to find the exact value of a trigonometric function without using a calculator.

In the exercise provided, we used the identity to convert \( 4 \sin 15^{\circ} \cos 15^{\circ} \) into a term involving \( \sin(30^{\circ}) \) by recognizing it as \( 2 \times 2 \sin(15^{\circ}) \cos(15^{\circ}) \). This is an excellent demonstration of how the double angle formula can simplify the process of evaluating trigonometric expressions.

Sine Function

The sine function is one of the fundamental trigonometric functions. It is used to relate the angle of a right triangle to the ratio of the opposite side over the hypotenuse. This function is periodic, with a period of \( 360^{\circ} \) (or \( 2\pi \) radians), and takes values between -1 and 1.

Most trigonometric problems involving the sine function require either calculating the value of the sine for a certain angle or manipulating expressions using sine identities. In our specific example, knowing the exact value of \( \sin(30^{\circ}) \) — which is \( \frac{1}{2} \) — allowed us to reach a conclusive solution efficiently.

• Sine of some key angles are well-known:
  • \( \sin(0^{\circ}) = 0 \)
  • \( \sin(30^{\circ}) = \frac{1}{2} \)
  • \( \sin(90^{\circ}) = 1 \)

Memorizing or understanding these key values is crucial as they frequently appear in exercises and solutions involving sine.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a common task in math courses, particularly in trigonometry and calculus. This process involves converting a complex trigonometric statement into a more straightforward or more recognizable form.

It often requires applying various trigonometric identities such as double angle formulas, Pythagorean identities, or sum and difference formulas. For the expression \( 4 \sin 15^{\circ} \cos 15^{\circ} \), the simplification process involved recognizing an opportunity to use the double angle formula.

Here are some steps to simplify trigonometric expressions effectively:

• Identify any recognizable trigonometric identity within the expression
• Substitute known identities to transform the expression
• Consolidate terms to reveal a single function or simplify to a numerical value
• Double-check by substituting back to ensure no incorrect manipulations happened

Simplification not only makes the expression easier but can also lead to meaningful insights about the problem, making it an essential skill for solving complex trigonometric problems.