Problem 12 - 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. \(A=45^{\circ}, B=270^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Recall the Angle Sum and Difference Formulas

The formulas for \(\sin(A - B)\) and \(\sin(A + B)\) are:\[sin(A - B) = \sin A \cos B - \cos A \sin B\]\[sin(A + B) = \sin A \cos B + \cos A \sin B\]

Step 2: Compute Trigonometric Values for A and B

Given \(A = 45^{\circ}\) and \(B = 270^{\circ}\), we need to calculate:\(\sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\).For \(270^{\circ}\): \(\sin 270^{\circ} = -1\) and \(\cos 270^{\circ} = 0\).

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

Substitute \(A = 45^{\circ}\) and \(B = 270^{\circ}\) into the formula:\[\sin(45^{\circ} - 270^{\circ}) = \sin 45^{\circ} \cdot \cos 270^{\circ} - \cos 45^{\circ} \cdot \sin 270^{\circ}\]\[= \frac{\sqrt{2}}{2} \cdot 0 - \frac{\sqrt{2}}{2} \cdot (-1)\]\[= \frac{\sqrt{2}}{2}\]

Step 4: Calculate \(\sin(A + B)\)

Now substitute \(A = 45^{\circ}\) and \(B = 270^{\circ}\) into the formula:\[\sin(45^{\circ} + 270^{\circ}) = \sin 45^{\circ} \cdot \cos 270^{\circ} + \cos 45^{\circ} \cdot \sin 270^{\circ}\]\[= \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{2}}{2} \cdot (-1)\]\[= -\frac{\sqrt{2}}{2}\]

Key Concepts

Angle Sum and Difference Formulas

The angle sum and difference formulas are powerful tools in trigonometry that help us find the sine and cosine of non-standard angles by using known values at standard angles. For the sine function, these formulas are:

• For the difference of two angles \(A\) and \(B\): \[\sin(A - B) = \sin A \cos B - \cos A \sin B\]
• For the sum of two angles \(A\) and \(B\):\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]

Using these equations, we can calculate new angles by breaking them down into parts we already understand. This is particularly useful for angles that don't come straight out of the unit circle but can be expressed as combinations of more familiar ones. In this exercise, we use these formulas to find exact values for \(\sin(A - B)\) and \(\sin(A + B)\).

Sine and Cosine Values

Understanding the values of sine and cosine at specific angles is critical to calculating sums and differences. For example, the angle \(45^{\circ}\) is a commonly used reference angle.

• The sine and cosine of \(45^{\circ}\) are both equal to \(\frac{\sqrt{2}}{2}\).

While angles like \(90^{\circ}\) and \(270^{\circ}\) have specific trigonometric properties:

• \(\sin(270^{\circ}) = -1\)
• \(\cos(270^{\circ}) = 0\)

These values are essential when plugging into the angle sum and difference formulas. Remembering these key values helps streamline your trigonometric calculations, making it quicker to solve problems on exams or assignments.

Trigonometric Calculations

Trigonometric calculations often involve breaking down complex expressions using known angle values and calculations. By applying the angle sum and difference formulas, we simplify the task of finding values for unfamiliar angles. Let’s consider calculating \(\sin(45^{\circ} - 270^{\circ})\) and \(\sin(45^{\circ} + 270^{\circ})\).First, substitute in the known sine and cosine values:

• For the difference formula: \[\sin(45^{\circ} - 270^{\circ}) = \frac{\sqrt{2}}{2} \cdot 0 - \frac{\sqrt{2}}{2} \cdot (-1) = \frac{\sqrt{2}}{2}\]
• For the sum formula: \[\sin(45^{\circ} + 270^{\circ}) = \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{2}}{2} \cdot (-1) = -\frac{\sqrt{2}}{2}\]

By leveraging these identities, you can solve complex trigonometric expressions by reducing them to simple, manageable parts.