Problem 110 - Precalculus

Question

For the following exercises, simplify to one trigonometric expression. $$ 4 \sin \left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{8}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt{2}\).

Step 1: Identify the Trigonometric Identity

Recognize that the expression given, \(4 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\), can be simplified using the double angle identity for sine. The double angle identity for sine is \(2 \sin(x) \cos(x) = \sin(2x)\).

Step 2: Apply the Double Angle Identity

Apply the identity \(2 \sin(x) \cos(x) = \sin(2x)\) to the expression \(4 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\) by first rewriting it as \(2 \times 2\sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\), which equals \(2 \sin\left(2 \times \frac{\pi}{8}\right)\).

Step 3: Simplify the Angle

Simplify the angle inside the sine function: \(2 \times \frac{\pi}{8} = \frac{\pi}{4}\). Thus, the expression \(2 \sin\left(\frac{\pi}{4}\right)\) is obtained.

Step 4: Calculate the Value of Sine

Recall that \(\sin\left(\frac{\pi}{4}\right)\) is a known value, which is \(\frac{\sqrt{2}}{2}\). Substitute this into the expression, resulting in \(2 \left(\frac{\sqrt{2}}{2}\right)\).

Step 5: Simplify the Final Expression

Simplify the multiplication \(2 \times \frac{\sqrt{2}}{2}\), which results in \(\sqrt{2}\).

Key Concepts

Double Angle Identity

The double angle identity for sine is an essential tool in trigonometry, allowing us to efficiently simplify expressions. It states that \(2 \sin(x) \cos(x) = \sin(2x)\). This identity transforms the product of a sine and cosine function into a single sine function with a doubled angle.

• It reduces complexity in expressions involving trigonometric functions.
• It facilitates calculations by converting difficult products into simpler summations or single functions.

Understanding the double angle identity is crucial when faced with expressions such as \(4 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\). Here, we apply it by splitting \(4\) as \(2 \times 2\sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\), hence transforming it into \(2 \sin\left(2 \times \frac{\pi}{8}\right)\). This simplification paves the way for easier calculations.

Angle Simplification

Simplifying angles is often an essential step after applying trigonometric identities. When using the double angle identity, the angle \(x\) is doubled, leading to potential further simplifications.

• For the expression \(2 \sin\left(2 \times \frac{\pi}{8}\right)\), simplify the angle \(\frac{2\pi}{8}\) to \(\frac{\pi}{4}\).
• This simplification uses basic arithmetic to transform fractions into more recognizable forms.

Divide or multiply the angle by integers when necessary to find simpler expressions. Keeping fractions manageable aids in both setting the stage for calculating known trigonometric values and facilitating a smooth simplification process.

Sine Function

The sine function is a fundamental component of trigonometry. It serves as a vital relation in many mathematical expressions. Knowing specific values of sine, such as \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), helps in swiftly solving trigonometric problems.

• The sine of an angle is equal to the length of the opposite side divided by the hypotenuse in a right triangle.
• Its graph is a wave that starts at zero, reaches a maximum of one, returns to zero, dips to negative one, and returns to zero once more over \(2\pi\).

In our context, after simplifying the angle, replacing \(\sin\left(\frac{\pi}{4}\right)\) with its known value continues the progression of transforming the original expression into a single, more manageable term.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into more straightforward components or known values. The goal is to achieve a minimal expression using identities and known values, simplifying computations or comparisons.

• Transform the expression using identities such as double angle or half angle for sine, cosine, and other trigonometric functions.
• Reduce fractions, perform multiplications, and apply known trigonometric values for simplification.

For \(4 \sin\left(\frac{\pi}{8}\right) \cos\left(\frac{\pi}{8}\right)\), through identity application and simplification: it resolves to \(\sqrt{2}\). Performing multiplications such as \(2 \times \frac{\sqrt{2}}{2}\) completes the transition from a complex trigonometric form to a simple, elegant result.