Problem 9 - Algebra 2 and Trigonometry

Question

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=30^{\circ}, B=90^{\circ}\)

Step-by-Step Solution

Answer

\(\sin(A-B) = -\frac{\sqrt{3}}{2}\) and \(\sin(A+B) = \frac{\sqrt{3}}{2}\).

Step 1: Understanding the Problem

In this exercise, we are given two angle measures: \(A = 30^{\circ}\) and \(B = 90^{\circ}\). We need to find the values of \(\sin(A-B)\) and \(\sin(A+B)\). These require the use of angle difference and sum identities for sine.

Step 2: Applying the Sine Difference Identity

The formula for the sine of difference is \(\sin(A-B) = \sin A \cos B - \cos A \sin B\). Substituting the given values:\[ \sin(30^{\circ} - 90^{\circ}) = \sin 30^{\circ} \cos 90^{\circ} - \cos 30^{\circ} \sin 90^{\circ} \]Given that \(\sin 30^{\circ} = \frac{1}{2}\), \(\cos 90^{\circ} = 0\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), and \(\sin 90^{\circ} = 1\), we find:\[ \sin(-60^{\circ}) = \frac{1}{2}(0) - \frac{\sqrt{3}}{2}(1) = -\frac{\sqrt{3}}{2} \]

Step 3: Applying the Sine Sum Identity

The formula for the sine of sum is \(\sin(A+B) = \sin A \cos B + \cos A \sin B\). Substituting the given values:\[ \sin(30^{\circ} + 90^{\circ}) = \sin 30^{\circ} \cos 90^{\circ} + \cos 30^{\circ} \sin 90^{\circ} \]Given that \(\sin 30^{\circ} = \frac{1}{2}\), \(\cos 90^{\circ} = 0\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), and \(\sin 90^{\circ} = 1\), we find:\[ \sin(120^{\circ}) = \frac{1}{2}(0) + \frac{\sqrt{3}}{2}(1) = \frac{\sqrt{3}}{2} \]

Key Concepts

Sine Difference Identity

The sine difference identity is a helpful formula for finding the sine of the difference between two angles. This identity is written as \( \sin(A-B) = \sin A \cos B - \cos A \sin B \). It allows us to calculate the exact value of \( \sin(A-B) \) using the sines and cosines of the individual angles.
For example, if we have two angles: \( A = 30^{\circ} \) and \( B = 90^{\circ} \):

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 90^{\circ} = 0 \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 90^{\circ} = 1 \)

By substituting these values into the identity:\[\sin(30^{\circ} - 90^{\circ}) = \frac{1}{2}(0) - \frac{\sqrt{3}}{2}(1) = -\frac{\sqrt{3}}{2} \]This formula is particularly useful when working with special angles like 30°, 60°, and 90°, where the exact trigonometric values are well-known.

Sine Sum Identity

The sine sum identity helps us find the sine of the sum of two angles. It is formulated as \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). This identity is key when dealing with angles where direct calculation might be complex, allowing us to leverage known values for simpler components.
Using the same angles \( A = 30^{\circ} \) and \( B = 90^{\circ} \):

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 90^{\circ} = 0 \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 90^{\circ} = 1 \)

Substituting these values, we have:\[\sin(30^{\circ} + 90^{\circ}) = \frac{1}{2}(0) + \frac{\sqrt{3}}{2}(1) = \frac{\sqrt{3}}{2} \]The sine sum identity makes tasks involving the sum of angles straightforward, especially when those angles are common reference angles like 30°, 45°, and 90°.

Special Angles

Special angles, such as 30°, 45°, 60°, and 90°, are angles whose trigonometric function values are generally memorized due to their simplicity and frequent occurrence in problems. These angles' sines and cosines are easy to recall and use. They form the basis for solving many trigonometric problems quickly.
For instance:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 90^{\circ} = 1 \)
• \( \cos 90^{\circ} = 0 \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)

By becoming familiar with these values, students can more easily tackle exercises involving trigonometric identities like sine difference and sum identities. These foundational angles are part of what makes solving trigonometry problems efficient and less of a memory exercise.