Problem 25 - Calculus with Concepts in Calculus

Question

Use the identities given in this section to compute the given value. $$ \sin \frac{7 \pi}{12}\left(\text { Hint }: \frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify Formula to Use

The expression \( \frac{7\pi}{12} \) can be rewritten as the sum \( \frac{\pi}{3} + \frac{\pi}{4} \). To find the sine of this sum, we use the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Substitute Known Values

Substitute \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \) into the formula. We have:\[ \sin \left( \frac{\pi}{3} \right) \cos \left( \frac{\pi}{4} \right) + \cos \left( \frac{\pi}{3} \right) \sin \left( \frac{\pi}{4} \right). \]

Step 3: Compute Trigonometric Values

Recall the values of sine and cosine for these angles:- \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)- \( \cos \frac{\pi}{3} = \frac{1}{2} \)- \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)- \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Step 4: Calculate Each Term

Compute the values for each part of the sine addition formula:1. \( \sin \frac{\pi}{3} \cos \frac{\pi}{4} = \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} \)2. \( \cos \frac{\pi}{3} \sin \frac{\pi}{4} = \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \)

Step 5: Add the Terms Together

Sum the results of the previous calculations:\[ \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}. \]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with problems involving angles and circles. They allow us to simplify expressions and solve complex trigonometric equations. One of the most common uses of identities is in the simplification of expressions, such as the sine addition formula. This particular identity, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), helps to break down problems where an angle is expressed as the sum of two other angles. By applying this identity, computations become more manageable, aiding in the evaluation of trigonometric functions without needing a calculator. These identities are particularly helpful in problems where direct calculation of an angle's sine, cosine, or tangent is complex or impossible. Understanding and memorizing these identities can make finding solutions much easier, as they provide a systematic way to resolve angle-related calculations.

Trigonometric Values

Trigonometric values for common angles are foundational to solving many types of trigonometric problems. Knowing the exact values for angles like \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) is crucial because these are frequently encountered in mathematics. For instance:

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

These values come from the unit circle and geometric properties of triangles. The ability to recall and use these values enables you to handle problems involving trigonometric identities more efficiently. Practice with these basic values reinforces a deeper understanding and facilitates recognition of patterns across different problems.

Angle Sum Formulae

The angle sum formulae are invaluable in trigonometry, providing a way to calculate the sine, cosine, or tangent of an angle expressed as the sum or difference of two other angles. The sine addition formula used in the original exercise is a great example of these formulae. It is structured as \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), applying directly when we rewrite an angle like \( \frac{7\pi}{12} \) as the sum \( \frac{\pi}{3} + \frac{\pi}{4} \). This breakdown helps simplify calculations because it reduces a potentially complex angle into simpler parts with known trigonometric values. Understanding how to use angle sum formulae effectively increases problem-solving efficiency and accuracy, as it allows for precise calculation of trigonometric functions without needing complicated computations or approximations. Mastery of these formulae is key to navigating more advanced trigonometric concepts seamlessly.