Problem 5 - 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. \(A=180^{\circ}, B=30^{\circ}\)}

Step-by-Step Solution

Answer

\( \sin(150^{\circ}) = \frac{1}{2} \) and \( \sin(210^{\circ}) = -\frac{1}{2} \).

Step 1: Understanding the Exercise

We need to find the exact values of \( \sin(A-B) \) and \( \sin(A+B) \). We are given \( A = 180^{\circ} \) and \( B = 30^{\circ} \). We will use the sine addition and subtraction formulas here.

Step 2: The Sine Subtraction Formula

The formula for \( \sin(A-B) \) is \( \sin(A)\cos(B) - \cos(A)\sin(B) \). We will calculate each trigonometric function using the given angles.

Step 3: Calculate \( \sin(180^{\circ}) \) and \( \cos(180^{\circ}) \)

From the unit circle, \( \sin(180^{\circ}) = 0 \) and \( \cos(180^{\circ}) = -1 \).

Step 4: Calculate \( \sin(30^{\circ}) \) and \( \cos(30^{\circ}) \)

From trigonometric ratios, \( \sin(30^{\circ}) = \frac{1}{2} \) and \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \).

Step 5: Apply the Sine Subtraction Formula

Substituting the values, \( \sin(180^{\circ} - 30^{\circ}) = \sin(180^{\circ})\cos(30^{\circ}) - \cos(180^{\circ})\sin(30^{\circ}) \). This simplifies to \( 0 \times \frac{\sqrt{3}}{2} - (-1) \times \frac{1}{2} \) which simplifies further to \( \frac{1}{2} \).

Step 6: The Sine Addition Formula

The formula for \( \sin(A+B) \) is \( \sin(A)\cos(B) + \cos(A)\sin(B) \). We'll use this with our calculated trigonometric values.

Step 7: Apply the Sine Addition Formula

Substituting into the formula: \( \sin(180^{\circ} + 30^{\circ}) = \sin(180^{\circ})\cos(30^{\circ}) + \cos(180^{\circ})\sin(30^{\circ}) \) becomes \( 0 \times \frac{\sqrt{3}}{2} + (-1) \times \frac{1}{2} \), which simplifies to \( -\frac{1}{2} \).

Key Concepts

Sine Addition Formula

The sine addition formula is a vital tool in trigonometry that helps us find the sine of the sum of two angles. It is given by the formula:

• \( \sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B) \)

This formula allows us to break down the sine of a sum into the components of each angle, making calculations more manageable. In practical terms, it helps to calculate when we have angles whose individual trigonometric values are known but not their sum. For example, in the case of given angles \(A = 180^{\circ}\) and \( B = 30^{\circ} \), applying this formula with their known sine and cosine values quickly gives you \( \sin(210^{\circ}) = -\frac{1}{2} \). This makes the sine addition formula a powerful technique in solving trigonometric problems.

Sine Subtraction Formula

Similar to the addition formula, the sine subtraction formula helps to determine the sine of the difference between two angles. The formula is:

• \( \sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B) \)

This formula is particularly useful when solving for the sine of an angle that can be expressed as the difference of two angles. For instance, in the original exercise with \(A = 180^{\circ}\) and \(B = 30^{\circ}\), the subtraction formula simplifies the solution for \( \sin(150^{\circ}) \), calculated as \( \frac{1}{2} \). By applying this formula, you can easily derive trigonometric functions without needing to memorize the sine values of many angles, therefore simplifying complex calculations.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. It beautifully ties together angles and trigonometric functions:

• Each point on the circle corresponds to an angle, measured from the positive x-axis.
• The x-coordinate of each point represents the cosine of the angle.
• The y-coordinate represents its sine.

For example, the unit circle helps to easily determine that \( \sin(180^{\circ}) = 0 \) and \( \cos(180^{\circ}) = -1 \), values crucial for processing the given calculations. Understanding the unit circle equips students with intuitive and quick ways to reason about angles and trigonometric values, reducing the need for rote memorization.

Trigonometric Ratios

Trigonometric ratios provide relationships between the angles and sides of triangles, which extend to solving problems involving sine, cosine, and tangent. In the context of the unit circle, they describe:

• Sine (\( \sin \)) - defined as the ratio of the opposite side to the hypotenuse in a right triangle. On the unit circle, it is the vertical component or y-value.
• Cosine (\( \cos \)) - the ratio of the adjacent side to the hypotenuse, which is the horizontal component or x-value on the unit circle.
• Tangent (\( \tan \)) - defined as the ratio of sine to cosine.

By understanding trigonometric ratios, you can effortlessly compute values like \( \sin(30^{\circ}) = \frac{1}{2} \) and \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \), essential for applying addition or subtraction formulas in trigonometry.