Problem 43 - Thomas Calculus

Question

Evaluate \(\sin \frac{7 \pi}{12}\) as \(\sin \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

\( \sin \frac{7\pi}{12} = \frac{\sqrt{2} + \sqrt{6}}{4} \).

Step 1: Recognize the Angle Sum Identity

We are given the angle \( \frac{7\pi}{12} \) and asked to evaluate \( \sin \frac{7\pi}{12} \) as an angle sum of \( \frac{\pi}{4} + \frac{\pi}{3} \). In mathematcis, the sine of the sum of two angles can be found using this identity: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Let \( A = \frac{\pi}{4} \) and \( B = \frac{\pi}{3} \).

Step 2: Find \( \sin A \) and \( \cos A \)

Substitute \( A = \frac{\pi}{4} \) into the sine and cosine functions. We know from the unit circle that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) and \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 3: Find \( \sin B \) and \( \cos B \)

Substitute \( B = \frac{\pi}{3} \) into the sine and cosine functions. From the unit circle, \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \).

Step 4: Apply the Angle Sum Identity

Using the angle sum identity \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), substitute all known values: \( \sin \frac{7\pi}{12} = \sin \left(\frac{\pi}{4} + \frac{\pi}{3}\right) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \).

Step 5: Simplify the Expression

Simplify the expression by calculating: \( \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{2}}{4} \) and \( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4} \). Add these results together: \( \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \).

Key Concepts

Angle Sum Identity

The angle sum identity is a powerful tool in trigonometry that helps simplify calculations involving the sine, cosine, and tangent functions. Specifically, it allows you to find the sine of a sum of two angles by using well-known values from the unit circle. This identity is expressed as:

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

The identity breaks down the calculation into simpler components by using the individual sine and cosine values of the angles you're working with. By isolating each part, you make the problem manageable.
In the given exercise, we use known angle measures, \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \), to find \( \sin \frac{7\pi}{12} \). The sine value of the combined angle \( \frac{7\pi}{12} \) is computed using these parts, showcasing how this identity streamlines solving complex angle problems. Understanding this identity is crucial for simplifying various trigonometric expressions when dealing with angle sums.

Unit Circle

The unit circle is a fundamental concept in trigonometry and is instrumental when dealing with trigonometric identities like the angle sum identity. It's a circle with a radius of 1 centered at the origin of the coordinate plane. This simple yet effective tool allows for easy calculation of sine, cosine, and tangent for various standard angles.
On the unit circle, each point along the circle corresponds to an angle \(\theta\), where the x-coordinate represents \(\cos \theta\) and the y-coordinate represents \(\sin \theta\). Understanding these coordinates helps in quickly finding specific trigonometric values. In this exercise, we use the unit circle to determine values for angles \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\).

• For \(\frac{\pi}{4}\): \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• For \(\frac{\pi}{3}\): \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\) and \(\cos \frac{\pi}{3} = \frac{1}{2}\)

These values form the building blocks for simplifying the expression \(\sin \left(\frac{\pi}{4} + \frac{\pi}{3} \right)\) and demonstrate the unit circle's practicality in solving trigonometric problems.

Simplifying Expressions

Simplifying trigonometric expressions is an essential skill in mathematics, and it is often achieved by using identities like the angle sum identity. In the given problem, once we apply the identity, the expression becomes:

• \( \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) + \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) \)

By handling each multiplication separately, you ensure clarity in your solution. Calculating each component:

• \( \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{2}}{4} \)
• \( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4} \)

Once these are calculated, the final step is to add them together to reach the simplified result:

• \( \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \)

This process not only helps in solving expressions but also strengthens your trigonometric understanding. Each step involved is a chance to practice breaking down and reconstructing expressions, which is invaluable in algebra and calculus.