Problem 10 - 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. \(A=90^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

Both \(\sin(A - B)\) and \(\sin(A + B)\) equal \(\frac{1}{2}\).

Step 1: Understand the Given Angles

We are given two angles: \(A = 90^{\circ}\) and \(B = 60^{\circ}\). Our task is to find the exact values of \(\sin(A - B)\) and \(\sin(A + B)\).

Step 2: Calculate \(\sin(A - B)\)

Substitute the angles into the \(\sin(A - B)\) formula: \(\sin(A - B) = \sin(90^{\circ} - 60^{\circ}) = \sin(30^{\circ})\). Since \(\sin(30^{\circ}) = \frac{1}{2}\), we have \(\sin(A - B) = \frac{1}{2}\).

Step 3: Calculate \(\sin(A + B)\)

Substitute the angles into the \(\sin(A + B)\) formula: \(\sin(A + B) = \sin(90^{\circ} + 60^{\circ}) = \sin(150^{\circ})\). Since \(\sin(150^{\circ}) = \sin(180^{\circ} - 30^{\circ}) = \sin(30^{\circ})\), and \(\sin(30^{\circ}) = \frac{1}{2}\), therefore \(\sin(A + B) = \frac{1}{2}\).

Key Concepts

Sine Addition Formula

The sine addition formula is a handy tool when you deal with precise angle measurements and trigonometric calculations. It helps you find the sine of the sum of two angles.The formula is expressed as:\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]Here's how it works in practical terms:

• When you know the values of angles \(A\) and \(B\), you can plug them into the formula.
• These values are typically in degrees or radians, and their sine and cosine values should be known or easily calculable.

For example, with our angles \(A = 90^{\circ}\) and \(B = 60^{\circ}\), the formula becomes:\[\sin(90^{\circ} + 60^{\circ}) = \sin 90^{\circ}\cos 60^{\circ} + \cos 90^{\circ} \sin 60^{\circ}\]Knowing that \(\sin 90^{\circ} = 1\), \(\cos 60^{\circ} = \frac{1}{2}\), \(\cos 90^{\circ} = 0\), and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), allows us to calculate:\[1 \times \frac{1}{2} + 0 \times \frac{\sqrt{3}}{2} = \frac{1}{2}\]Thus, \(\sin(150^{\circ}) = \frac{1}{2}\). This example shows how understanding this formula can simplify complex calculations.

Sine Subtraction Formula

Understanding the sine subtraction formula is crucial when you need to determine the sine of the difference between two angles.The formula is as follows:\[\sin(A - B) = \sin A \cos B - \cos A \sin B\]Using this formula makes evaluating trigonometric expressions involving subtractions more manageable.

• It's important to remember the angle properties and their respective sine and cosine values to apply this formula effectively.
• Tackle it like a puzzle: fill in the known values and solve.

As with the sine addition example, plug in \(A = 90^{\circ}\) and \(B = 60^{\circ}\):\[\sin(90^{\circ} - 60^{\circ}) = \sin 90^{\circ}\cos 60^{\circ} - \cos 90^{\circ}\sin 60^{\circ}\]Since \(\sin 90^{\circ} = 1\), \(\cos 60^{\circ} = \frac{1}{2}\), \(\cos 90^{\circ} = 0\), and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), the calculation goes as follows:\[1 \times \frac{1}{2} - 0 \times \frac{\sqrt{3}}{2} = \frac{1}{2}\]Hence, \(\sin(30^{\circ}) = \frac{1}{2}\). With practice, this formula becomes a straightforward part of solving trigonometric problems.

Angle Measurements

In trigonometry, angle measurements are essential because they help determine the specific trigonometric values needed for various calculations. It's important to understand both degrees and radians as units of measurement since each has its role in mathematics.Degrees are commonly used in everyday life scenarios, like navigation or in a school setting, whereas radians are predominantly found in higher level mathematics and physics due to their natural mathematical properties.

• 1 full circle = \(360^{\circ}\) or \(2\pi\) radians
• Right angle = \(90^{\circ}\) or \(\frac{\pi}{2}\) radians
• Half circle = \(180^{\circ}\) or \(\pi\) radians

When dealing with trigonometric identities, understanding how to convert between these units becomes vital. Conversion between degrees and radians is done using the formula:\[\text{Radians} = \left( \frac{\pi}{180} \right) \times \text{Degrees}\]And vice versa:\[\text{Degrees} = \left( \frac{180}{\pi} \right) \times \text{Radians}\]In our specific exercise, angles were given in degrees. But if needed, they could easily be converted, ensuring calculations remain accurate regardless of the unit. Keeping these conversion techniques handy helps streamline solving trigonometric identities.