Problem 43 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Angle Sum

We are given \[ \sin \frac{7\pi}{12} \]which can be expressed as \[ \sin \left(\frac{\pi}{4} + \frac{\pi}{3}\right) \]This is the sum of two common angles, \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), which makes it a candidate for using the angle sum identity for sine.

Step 2: Recall the Sine Angle Sum Identity

The sine angle sum identity is given by:\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]We will use this identity where \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{3} \).

Step 3: Substitute and Simplify

Substitute \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{3} \) into the sine angle sum identity:\[ \sin\left(\frac{\pi}{4} + \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{3}\right) \]Recall the trigonometric values:- \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)- \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)- \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

Step 4: Calculate the Expression

Calculate the expression using the known values:\[ \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \]This simplifies to:\[ \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \]Combine the fractions:\[ \frac{\sqrt{2} + \sqrt{6}}{4} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that relate the angles and sides of triangles. They help simplify complex expressions and solve trigonometric equations easily.
The most common trigonometric identities include:

• Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal identities, like \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Angle sum and difference identities, which include the sine and cosine angle sum formulas.

These identities are essential for evaluating expressions involving trigonometric functions and for proving other mathematical statements. Understanding and mastering trigonometric identities enables you to solve a wide variety of mathematical problems.

Angle Addition Formula

The angle addition formula plays a critical role in trigonometry, particularly when dealing with the sum or difference of two angles. For sine, the formula is:\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]This identity allows us to find the sine of an angle that is expressed as a sum of two standard angles, like \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \).
The formula helps break down complex angles into simpler components, which are often easier to evaluate using known trigonometric values.

• If you know the sine and cosine of angles \( a \) and \( b \), you can determine the sine of their combined angle by applying this formula.
• It is particularly useful in real-world applications, where measurements don't often align with standard angles, requiring adjustments using these identities.

By familiarizing yourself with the angle addition formula, you can tackle a broader range of problems in trigonometry.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves calculating the exact value of expressions involving trigonometric functions. This process is crucial when dealing with angles that don't have immediate, simple trigonometric values.
In the given example, \( \sin \frac{7\pi}{12} \) is restructured as \( \sin(\frac{\pi}{4} + \frac{\pi}{3}) \). By using the angle addition formula:\[ \sin\left(\frac{\pi}{4} + \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{3}\right) \]we can substitute the known values:

• \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
• \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
• \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
• \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)

Upon simplifying the expression, the result is \( \frac{\sqrt{2} + \sqrt{6}}{4} \). Understanding how to evaluate these expressions is vital for achieving precise trigonometric values in diverse mathematical contexts.