Problem 13 - 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. \(A=30^{\circ}, B=270^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Compute sin(A-B)

\( \sin(30^\circ - 270^\circ) = \sin(-240^\circ) = -\sin(240^\circ) = -(-\frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{2} \).
Wait, let us use the formula: \( \sin(A-B) = \sin A \cos B - \cos A \sin B \).
\( = \sin 30^\circ \cos 270^\circ - \cos 30^\circ \sin 270^\circ = \frac{1}{2}(0) - \frac{\sqrt{3}}{2}(-1) = \frac{\sqrt{3}}{2} \).

Step 2: Compute sin(A+B)

\( \sin(30^\circ + 270^\circ) = \sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2} \).
Using the formula: \( \sin A \cos B + \cos A \sin B = \frac{1}{2}(0) + \frac{\sqrt{3}}{2}(-1) = -\frac{\sqrt{3}}{2} \).

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It relates to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In simpler terms, when you look at an angle in a right triangle, the sine of that angle produces a ratio that helps describe the size of the triangle’s features. To make it easier to understand, think of it this way:

• If you have a right-angled triangle and know the angle, the sine provides a way to determine the relationship between two sides of that triangle.
• Using a scientific calculator or the unit circle, you can find that this ratio remains constant at a given angle.

The sine function is often denoted as \( \sin \theta \), where \( \theta \) is the angle in question. Understanding the sine function’s output helps with a deeper understanding of wave behaviors, circular motion, and geometrical properties of shapes.

Unit Circle

The unit circle is like a magic tool for trigonometry, giving you the sine, cosine, and tangent of all angles. It's a circle with a radius of 1, centered at the origin of the coordinate plane. This property makes calculations clean and straightforward.Here's why the unit circle is powerful:

• Each point on the circle corresponds to an angle with respect to the positive x-axis.
• The x-coordinate of a point on the unit circle is the cosine of the angle, and the y-coordinate is the sine of the angle.
• This means every angle can be tied directly to a specific point, simplifying finding the sine or cosine of various angles.

For example, at \( 30^{\circ} \), the coordinates are \( \left( \sqrt{3}/2, 1/2 \right) \), giving \( \sin(30^{\circ}) = 1/2 \). At \( 270^{\circ} \), aligned along the y-axis, its coordinates shift, showing \( \sin(270^{\circ}) = -1 \). Having a mental model of this circle aids greatly in working out trigonometric problems.

Angle Subtraction and Addition Formulas

When working with angles, you will often need to find trigonometric values for the sum or difference of two angles. This is where the angle subtraction and addition formulas become very handy.For the sine function, these formulas are:

• Angle Addition Formula for Sine: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
• Angle Subtraction Formula for Sine: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]

These formulas allow you to calculate the sine of composite angles. For example, using these formulas, you can compute:

• \( \sin(A + B) \) for \( A = 30^{\circ} \) and \( B = 270^{\circ} \); plug in the values from the unit circle into the formula to find exact values.
• Similarly, \( \sin(A - B) \), giving another angle abbreviation result.

Mastering these concepts can significantly simplify complex trigonometric problems, making them more approachable and less time-consuming.