Problem 9 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$\sin \frac{19 \pi}{12}$$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{2} + \sqrt{6}}{4} \).

Step 1: Identify Known Angles

The goal is to express the angle \( \frac{19 \pi}{12} \) in terms of angles for which we know the exact trigonometric values. Two common angles are \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \), which are \( 45^\circ \) and \( 60^\circ \) respectively. Notice that \( \frac{19 \pi}{12} = \frac{6 \pi}{12} + \frac{8 \pi}{12} + \frac{5 \pi}{12} = \pi + \frac{2 \pi}{3} + \frac{\pi}{4} \) simplifies to \( \frac{3 \pi}{4} + \frac{\pi}{3} \).

Step 2: Express Value Using Known Angles

We use the identity for sine of a sum, \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Let \( A = \frac{\pi}{4} \) and \( B = \frac{\pi}{3} \). Then we need \( \sin \frac{\pi}{4} \), \( \cos \frac{\pi}{4} \), \( \sin \frac{\pi}{3} \), and \( \cos \frac{\pi}{3} \).

Step 3: Use Known Sine and Cosine Values

From trigonometric values: \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), \( \cos \frac{\pi}{3} = \frac{1}{2} \). Substitute these into the identity.

Step 4: Apply Addition Formula

\[ \sin \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \sin \frac{\pi}{4} \cos \frac{\pi}{3} + \cos \frac{\pi}{4} \sin \frac{\pi}{3} \] \[ = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \] \[ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \].

Step 5: Simplify Expression

Combine the terms over a common denominator: \[ \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4} \].

Key Concepts

Addition Formula

The addition formula in trigonometry is a powerful tool that allows us to find the sine or cosine of the sum of two angles. These formulas come in handy when dealing with angles that don't have direct known values. Here's what the formula looks like for sine:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

We use the addition formula when we can express a difficult angle as the sum of two familiar angles. This way, we can break down the operation into simpler steps using known trigonometric values. In our exercise, we expressed the angle \( \frac{19\pi}{12} \) as \( \frac{\pi}{4} + \frac{\pi}{3} \). By doing this, we are able to apply the formula directly. First, identify the angles \( A \) and \( B \) and their corresponding sine and cosine values. Then, substitute these values into the formula and simplify the expression.
By learning and applying the addition formula, you're opening a doorway to solve complex trigonometric expressions more easily.

Subtraction Formula

While our problem focused on the addition formula, the subtraction formula works similarly and is equally useful in trigonometry. It is particularly helpful when you need to find the sine or cosine of the difference between two angles. The subtraction formulas are as follows:

• For sine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
• For cosine: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas allow great flexibility in trigonometric computations, making it easier to evaluate angles that might not ordinarily have simple values. By using known angles with exact trigonometric values, you can handle complex expressions effectively.
Understanding both addition and subtraction formulas enriches your toolkit for tackling various trigonometric problems with confidence.

Exact Trigonometric Values

When solving trigonometric problems, having a firm grasp on the exact trigonometric values for common angles is essential. These values serve as foundational building blocks in more complex calculations.
Here's a quick review of exact trigonometric values for key angles:

• For \( 45^\circ \) or \( \frac{\pi}{4} \):
  \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• For \( 60^\circ \) or \( \frac{\pi}{3} \):
  \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \)

These exact values allow you to compute expressions without a calculator, by breaking down angles using addition and subtraction formulas. Utilizing a strong understanding of these values, you can approach seemingly difficult trigonometric equations with assurance and accuracy.