Problem 46 - Algebra 2

Question

Find the exact value of each expression. \(\sin 15^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Step 1: Recall Angle Sum Identity

The angle sum identity for sine is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We can express \(15^{\circ}\) as \(45^{\circ} - 30^{\circ}\).

Step 2: Apply the Identity to Express \(\sin 15^{\circ}\)

Using the identity, \( \sin 15^{\circ} = \sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ} \).

Step 3: Substitute Exact Values

Now substitute the values: \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 4: Simplify the Expression

Plugging in the values, we get \( \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \). This simplifies to \( \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \).

Step 5: Combine Terms

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

Key Concepts

Angle Sum Identity

The angle sum identity is a fundamental concept in trigonometry. It provides a way to find the sine, cosine, or tangent of a sum or difference of two angles. Specifically, for sine, the angle sum identity is given by:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• This identity helps simplify complex trigonometric expressions by breaking them down into components of known angles.

In the problem at hand, we apply this identity to \(15^{\circ}\) by expressing it as \(45^{\circ} - 30^{\circ}\). This allows us to use the known values of trigonometric functions for these common angles, making the computation precise and straightforward.

Exact Values of Trigonometric Functions

Trigonometric functions for common angles like \(30^{\circ}, 45^{\circ},\) and \(60^{\circ}\) have well-established exact values. For instance:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 30^{\circ} = \frac{1}{2} \)

These exact values are derived from special right triangles. For example, the values for \(30^{\circ}\) and \(60^{\circ}\) come from the ratios within an equilateral triangle split in half. Memorizing these values is crucial for efficiently solving trigonometric problems without the need for calculators.

Sine Function

The sine function is one of the primary trigonometric functions. It relates a given angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. Not only is sine important in geometry, but it's also vital in various real-world applications, such as physics and engineering.The sine wave is periodic, meaning it repeats its values in regular intervals of \(360^{\circ}\) or \(2\pi\). Understanding the properties of the sine function enables students to grasp concepts like amplitude, frequency, and phase shift—essential for understanding wave behavior in sciences.

Simplification of Expressions

Simplification of expressions is key in mathematics, particularly in dealing with trigonometric identities and complex calculations. By breaking down expressions into their simplest forms, we can more easily compare, analyze, and compute values.In our exercise, after applying the angle sum identity and substituting the exact values of trigonometric functions, we arrive at an expression that looks like this:

• \( \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \)

The goal is to simplify. First, carry out the multiplications, yielding \( \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \). Combining these into a single fraction results in the neat expression \( \frac{\sqrt{6} - \sqrt{2}}{4} \), showing the exact value of \(\sin 15^{\circ}\) in its simplest form. Simplification not only makes results easier to read but also ensures accuracy in further calculations.