Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \left(-\frac{5 \pi}{12}\right)$$

Step-by-Step Solution

Answer

\( \sin\left(-\frac{5\pi}{12}\right) = \frac{-\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Identify the Identity to Use

To find the exact value of \( \sin(-\frac{5\pi}{12}) \), we utilize the sine angle addition identity because the angle \( -\frac{5\pi}{12} \) can be expressed as a sum or difference of angles with known sine values. The sine addition identity is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Express the Angle

The angle \( -\frac{5\pi}{12} \) can be rewritten using known angles like: \( -\frac{5\pi}{12} = -\frac{\pi}{4} - \frac{\pi}{6} \). Both angles \( -\frac{\pi}{4} \) and \( -\frac{\pi}{6} \) have known sine and cosine values, which makes them suitable for use in the identity.

Step 3: Apply the Sine Angle Addition Identity

Applying the identity for sine, we get:\[sin\left(-\frac{5\pi}{12}\right) = \sin\left(-\frac{\pi}{4} - \frac{\pi}{6}\right) = \sin\left(-\frac{\pi}{4}\right)\cos\left(-\frac{\pi}{6}\right) - \cos\left(-\frac{\pi}{4}\right)\sin\left(-\frac{\pi}{6}\right).\]

Step 4: Substitute Known Values

Substitute the known values:- \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)- \( \cos\left(-\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)- \( \cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)- \( \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \).These values convert the expression to:\[-\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \left(-\frac{1}{2}\right)\]

Step 5: Simplify the Expression

Calculate the product in each term:\[-\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{-\sqrt{6} + \sqrt{2}}{4}.\]

Step 6: Conclusion

Thus, the exact value of \( \sin\left(-\frac{5\pi}{12}\right) \) is \( \frac{-\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Sine Angle Addition Identity

The Sine Angle Addition Identity is a fundamental tool in trigonometry. It helps us find the sine of an angle that can be expressed as the sum or difference of two other known angles. The identity is written as:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

This formula is particularly useful when dealing with angles that are not standard or easily recognizable from trigonometric charts.
In the exercise, we applied the Sine Angle Addition Identity to \( \sin\left(-\frac{5\pi}{12}\right) \). By breaking down \( -\frac{5\pi}{12} \) into simpler angles like \( -\frac{\pi}{4} \) and \( -\frac{\pi}{6} \), both of which have known sine and cosine values, we were able to plug them into the identity, making our calculations more straightforward.

Exact Trigonometric Values

Trigonometric values such as sine and cosine are often memorized for standard angles like \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\). Knowing these values by heart simplifies the process when solving trigonometry problems.
For instance:

• \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
• \( \cos\left(-\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
• \( \cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)
• \( \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \)

Substituting these exact values into the trigonometric identities results in accurate solutions without the need for a calculator. This ability to recall exact trigonometric values is a critical skill in both academic and practical applications of mathematics.

Simplifying Expressions

Simplifying trigonometric expressions is an essential skill. It involves reducing complex mathematical expressions into a simpler, more digestible form.
In our exercise, after substituting known trigonometric values, we were left with the expression:

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

Calculating each product step by step, we derived:

• \(-\frac{\sqrt{6}}{4}\) from the first term, and \(+ \frac{\sqrt{2}}{4}\) from the second term.

Combining these results simplified to the final result:

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

Taking each part methodically ensures clarity and accuracy, which is essential for mastering trigonometry problems.