Problem 109 - Precalculus

Question

For the following exercises, simplify to one trigonometric expression. $$ 2 \sin \left(\frac{\pi}{4}\right) 2 \cos \left(\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Identify Trigonometric Product

We start with the expression \( 2 \sin\left(\frac{\pi}{4}\right) 2 \cos\left(\frac{\pi}{4}\right) \). Recognize that this is a product of \( 2 \sin A \cos A \).

Step 2: Use Double-Angle Identity

The double-angle identity for sine states that \( \sin(2A) = 2 \sin A \cos A \). Therefore, we recognize the expression \( 2 \sin\left(\frac{\pi}{4}\right) 2 \cos\left(\frac{\pi}{4}\right) \) is equivalent to \( 2 \sin\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{4}\right) = \sin\left(2 \times \frac{\pi}{4}\right) \).

Step 3: Simplify the Angle

Calculate \( 2 \times \frac{\pi}{4} = \frac{\pi}{2} \). Therefore, the expression simplifies to \( \sin\left(\frac{\pi}{2}\right) \).

Step 4: Evaluate the Sine Function

The sine of \( \frac{\pi}{2} \) is known to be 1. Therefore, \( \sin\left(\frac{\pi}{2}\right) = 1 \). Hence, the simplified expression is 1.

Key Concepts

Double-Angle Identity

The Double-Angle Identity is a key tool in trigonometry that helps simplify expressions involving trigonometric functions by combining two angles into a single angle. This identity is especially useful when dealing with products of sine and cosine functions.
In the case of sine, the double-angle identity states:

• \( \sin(2A) = 2 \sin A \cos A \)

This identity allows us to express the product \( 2 \sin A \cos A \) as \( \sin(2A) \). This can be very handy in situations like our exercise, where we started with a product of sine and cosine, allowing us to convert it into a simpler sine expression of a doubled angle. Understanding and applying the double-angle identity can greatly simplify complex trigonometric calculations.

Sine Function

The sine function is one of the fundamental trigonometric functions. It is used to measure the vertical component of a point on the unit circle, which is defined as the circle with a radius of one centered at the origin of a coordinate plane. In practical terms, for a given angle, the sine function gives the ratio of the opposite side to the hypotenuse in a right-angled triangle.
Knowing specific values of the sine function, like \( \sin(\frac{\pi}{2}) \), is crucial for solving trigonometric problems. These key values are often derived from special angles and enable us to quickly evaluate trigonometric expressions. For example, \( \sin(\frac{\pi}{2}) = 1 \) reflects the fact that at \( \frac{\pi}{2} \), the point on the unit circle is directly above the origin, exactly one unit away vertically.

Simplification Techniques

Simplification techniques in trigonometry often involve recognizing patterns, such as identities, that can transform complex expressions into simpler ones. The initial step in applying these techniques is to identify components of the expression that fit known identities.
In the exercise, the expression \( 2 \sin\left(\frac{\pi}{4}\right) 2 \cos\left(\frac{\pi}{4}\right) \) was initially broken down using the recognition of a pattern typical of the double-angle identity: \( 2 \sin A \cos A \). By applying the double-angle identity, we were able to reduce the original expression to \( \sin\left(\frac{\pi}{2}\right) \). After noting the value of \( \sin\left(\frac{\pi}{2}\right) \) as \( 1 \), the expression was completely simplified.
This kind of pattern recognition is the essence of simplification techniques, providing a pathway to make dealing with trigonometric expressions much more manageable.