Problem 4 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin 15^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \sin 15^{\circ} \) is \( \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1: Recognize the Formula Needed

To find \( \sin 15^{\circ} \), we can use the sine subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). We notice that \( 15^{\circ} \) can be expressed as \( 45^{\circ} - 30^{\circ} \). Therefore, we will use the identity for \( \sin(45^{\circ} - 30^{\circ}) \).

Step 2: Apply the Sine Subtraction Formula

Plug \( a = 45^{\circ} \) and \( b = 30^{\circ} \) into the sine subtraction formula. It becomes:\[\sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ}.\]

Step 3: Substitute Known Values

Use known trigonometric values: \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \). Substitute these values into the formula:\[\sin 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right).\]

Step 4: Perform the Calculations

Calculate the expression:\[\sin 15^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}.\]Combine the terms:\[\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}.\]This is the exact value of \( \sin 15^{\circ} \).

Key Concepts

Sine Subtraction Formula

The sine subtraction formula is a fundamental tool used in trigonometry to find the sine of the difference between two angles, say \( a \) and \( b \). Formally, the formula is expressed as: \[\sin(a - b) = \sin a \cos b - \cos a \sin b \]This equation allows us to break down a seemingly complex sine function into simpler components.

• It uses the sine and cosine values of the individual angles \( a \) and \( b \).
• It provides a straightforward way to determine exact values without resorting to a calculator.

For example, in the calculation of \( \sin 15^{\circ} \), the angle \( 15^{\circ} \) is expressed as the difference of \( 45^{\circ} - 30^{\circ} \). By knowing the sine and cosine of \( 45^{\circ} \) and \( 30^{\circ} \), we can use this formula effectively. Thus, the sine subtraction formula is especially helpful in situations where one needs to find exact trigonometric values quickly.

Exact Trigonometric Values

Exact trigonometric values are crucial when working with formulas like the sine subtraction formula. They allow us to compute the exact sine, cosine, and tangent of many common angles.

• Some commonly used angles include \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \).
• Remembering these values can greatly reduce calculation time.

For instance, we have:\[\sin 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \]\[\cos 30^{\circ} = \frac{\sqrt{3}}{2}, \quad \sin 30^{\circ} = \frac{1}{2}\]These values are derived from the properties of special triangles:

• The 45-45-90 triangle.
• The 30-60-90 triangle.

Knowing these provides a foundation for calculating expressions like \( \sin 15^{\circ} \), helping us to find its exact value without having to approximate the result.

Angle Subtraction

Angle subtraction is a technique used in trigonometry to simplify the evaluation of trigonometric expressions. By expressing a given angle as the difference between two known angles, we can apply trigonometric identities more easily. Typically, this step is crucial for leveraging formulas like the sine subtraction formula. Here’s why it works so effectively:

• It transforms a difficult-to-calculate angle into a combination of angles with known trigonometric values.
• It simplifies the problem and reduces the potential for calculation errors.

For instance, in finding \( \sin 15^{\circ} \), we broke it down into \( 45^{\circ} \) and \( 30^{\circ} \), taking advantage of the sine subtraction formula. Using this technique is a powerful way to make complex trigonometric problems manageable, while also enriching your understanding and appreciation of trigonometry’s structure and reliance on foundational identities.