Problem 14 - 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. \(A=360^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

\( \sin(300^{\circ}) = -\frac{\sqrt{3}}{2} \) and \( \sin(420^{\circ}) = \frac{\sqrt{3}}{2} \).

Step 1: Understand the Problem

We need to find the exact values of \( \sin(A - B) \) and \( \sin(A + B) \) for the angles \( A = 360^{\circ} \) and \( B = 60^{\circ} \). These require the use of the sine subtraction and addition formulas.

Step 2: Apply the Sine Subtraction Formula

The sine of the difference of two angles is given by the formula: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Substitute \( A = 360^{\circ} \) and \( B = 60^{\circ} \) into the formula.

Step 3: Calculate \( \sin(360^{\circ}) \) and \( \cos(60^{\circ}) \)

\( \sin(360^{\circ}) = 0 \) because 360 degrees is a full circle returning to the origin. \( \cos(60^{\circ}) = \frac{1}{2} \) by trigonometric values of special angles.

Step 4: Calculate \( \cos(360^{\circ}) \) and \( \sin(60^{\circ}) \)

\( \cos(360^{\circ}) = 1 \) as it corresponds to the same direction as 0 degrees along the x-axis. \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \), a standard trigonometric value.

Step 5: Substitute and Solve \( \sin(A - B) \)

Using the values found, substitute into the formula: \[ \sin(360^{\circ} - 60^{\circ}) = 0 \times \frac{1}{2} - 1 \times \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} \].

Step 6: Apply the Sine Addition Formula

The sine of the sum of two angles is given by the formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Substitute \( A = 360^{\circ} \) and \( B = 60^{\circ} \) into this formula.

Step 7: Substitute and Solve \( \sin(A + B) \)

Using the trigonometric values: \[ \sin(360^{\circ} + 60^{\circ}) = 0 \times \frac{1}{2} + 1 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \].

Key Concepts

Sine Addition Formula

The Sine Addition Formula is an essential trigonometric identity that helps us find the sine of the sum of two angles. This formula is expressed as \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]. This means, when adding two angles together, you multiply the sine of one angle by the cosine of the other, and then do the same operation in reverse before adding these results together. This formula is incredibly useful when angles aren't directly derivable from the unit circle and we need this relationship to break them down into known values.

• For example, you can determine the sine of the sum of angles like 360° and 60° using their known sine and cosine values.
• By substituting these values into the formula, you acquire \[ \sin(360^{\circ} + 60^{\circ}) = 0 \cdot \frac{1}{2} + 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \].

With practice, utilizing the sine addition formula becomes second nature and greatly simplifies evaluating complex trigonometric expressions.

Sine Subtraction Formula

The Sine Subtraction Formula serves a similar function as its addition counterpart but is focused on finding the sine of the difference between two angles. The formula is given by: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]. Here, we take the sine of the first angle and multiply it with the cosine of the second angle, then perform the reverse operation and subtract the two results.

• This formula is particularly useful in solving problems where you need to break down a complicated angle into more simplistic terms that are easier to compute.
• Such as calculating \[ \sin(360^{\circ} - 60^{\circ}) \], which results in \[ 0 \cdot \frac{1}{2} - 1 \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} \].

Using the sine subtraction formula allows for efficient calculation and provides clarity when working with trigonometric functions involving non-standard angles.

Special Angles in Trigonometry

Special angles in trigonometry, like 0°, 30°, 45°, 60°, and 90°, have specific sine and cosine values that serve as building blocks for solving complex trigonometric problems. Understanding these angles and their values is crucial for applying trigonometric identities and formulas effectively.

• The angle 360° corresponds to completing a full circle on the unit circle, making its sine 0 and cosine 1, reverting back to the starting point.
• The angle 60°, however, yields the sine value \( \frac{\sqrt{3}}{2} \) and the cosine value \( \frac{1}{2} \), which are derived from the unit circle and right-angled triangles.

These consistent patterns make it easier to apply formulas like sine addition and subtraction. By referring back to these special angles, you can deduce the trigonometric outputs rapidly, improving problem-solving efficiency and accuracy in mathematics. Therefore, practicing these helps strengthen your foundational understanding and application of trigonometry.