Problem 2 - Algebra and Trigonometry

Question

\(1-12\) : 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

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

Step 1: Identify Trigonometric Identities

The exercise requires us to use an addition or subtraction formula to find the exact value of \( \sin 15^{\circ} \). Relevant trigonometric identities to consider are the angle addition and subtraction formulas.\[\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta\]To write 15° as a sum or difference of angles, we can choose 15° as \(45^{\circ} - 30^{\circ}\).

Step 2: Apply the Sine Subtraction Formula

Use the subtraction formula for sine with \( \alpha = 45^{\circ} \) and \( \beta = 30^{\circ} \). Applying the formula gives: \[\sin(15^{\circ}) = \sin(45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ}\]

Step 3: Substitute Known Values

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

Step 4: Simplify the Expression

Simplify the products and combine the terms: \[\sin(15^{\circ}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\] This is the exact value of \( \sin(15^{\circ}) \).

Key Concepts

Angle Addition Formula

The angle addition formula is extremely useful in trigonometry because it allows us to calculate the trigonometric functions of non-standard angles. For the sine function, the addition formula is expressed as:\[\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\]This formula can be a bit tricky at first, but it breaks down the problem by using angles whose exact trigonometric values we already know. Let's see how it works:

• Pick two angles \( \alpha \) and \( \beta \) to add together to create the angle you're interested in.
• Use known sine and cosine values for these angles to fill in the formula.

The angle addition formula allows us to obtain exact trigonometric values without using a calculator, an essential skill for mathematics and science problems involving precision and exact values. It's a powerful technique, especially when dealing with angles multiples of 15° or 30°, as their sums often consist of more familiar angles like 30°, 45°, or 60°, where exact values are known.

Angle Subtraction Formula

The angle subtraction formula is similar to the angle addition formula, but it involves subtracting one angle from another. For the sine function, the angle subtraction formula is:\[\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta\]This formula is essential for finding the exact values of sine for angles like 15° or 75°, which are not available directly in common trigonometric tables. To use the angle subtraction formula effectively:

• Select angles \( \alpha \) and \( \beta \) that you know, which subtract to give the desired angle.
• Substitute their known trigonometric values into the formula.
• Simplify the result to find the exact value.

Let's take \( \sin(15^{\circ}) \) as an example:

• Express 15° as 45° - 30°.
• Using the formula, compute \( \sin(45^{\circ} - 30^{\circ}) \).

This method helps you find values without estimation errors that arise from using calculators.

Exact Values of Trigonometric Functions

To solve trigonometric problems like finding \( \sin(15^{\circ}) \) using the angle subtraction formula, it's crucial to know the exact values of some common angles. Here are a few examples with their trigonometric functions:

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

Knowing these exact values makes it possible to fill in the formulas and find the sine or cosine of other angles by combining or splitting them. These exact values are fundamental for simplifying trigonometric expressions in math, physics, and engineering. When working through problems, always check if you can express an angle as a sum or difference of these common angles, as it often simplifies the process and leads directly to an exact solution. By understanding and memorizing these exact values, you'll gain a significant advantage in solving trigonometric problems without a calculator!