Problem 9 - Algebra and Trigonometry

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 of \( \sin \frac{19\pi}{12} \) is \(-\frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Understand the Identity

To solve this problem, we'll use the angle addition formula for sine, which is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We can also consider the angle subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Our goal is to express \( \frac{19\pi}{12} \) as a sum or difference of angles for which the sine and cosine are known.

Step 2: Break Down the Angle

We need to express \( \frac{19\pi}{12} \) as a sum or difference of angles whose trigonometric functions we know. Recognize that \( \frac{19\pi}{12} = \pi + \frac{7\pi}{12}\). This is equivalent to \( 180^\circ + 105^\circ \). We know \( \sin 105^\circ \) can be found by using \( \sin(60^\circ + 45^\circ) \).

Step 3: Apply the Addition Formula

To find \( \sin(60^\circ + 45^\circ) \) using the angle addition formula: \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \); \( \cos(60^\circ) = \frac{1}{2} \) and \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \). Substitute these into the formula:\[\sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2}\].

Step 4: Simplify

Simplify the expression obtained from the angle addition:\[\sin(60^\circ + 45^\circ) = \frac{\sqrt{3} \cdot \sqrt{2}}{4} + \frac{1 \cdot \sqrt{2}}{4} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.\] Since \( \sin\left(\pi + \theta \right) = -\sin\theta \),\[\sin\left(\pi + \frac{7\pi}{12}\right) = -\sin\left(\frac{7\pi}{12}\right)\].

Step 5: Final Result

Therefore, the exact value of \( \sin \frac{19\pi}{12} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Addition Formula for Sine

The addition formula for sine is a helpful tool for finding the sine of the sum of two angles. It is expressed as \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). This formula allows us to decompose a more complex angle into simpler components that are easier to work with.

• First, identify two angles \(a\) and \(b\) such that their sum equals the given angle.
• Calculate the sine and cosine of each component angle.
• Insert these values into the formula to find the sine of the original angle.

By utilizing this method, we simplify the computation of trigonometric values, especially when the exact values of the individual angles are known. Ensuring clear understanding of this formula enables solving otherwise complex trigonometric challenges with ease.

Subtraction Formula for Sine

Sometimes, we need to find the sine of a difference between two angles. This is where the subtraction formula for sine comes in handy. It is written as \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

Understanding how this formula works is crucial for solving problems where an angle is expressed as the difference of two simpler angles. The process involves these steps:

• Select angles \(a\) and \(b\) such that their difference equals the provided angle.
• Determine the sine and cosine of each angle.
• Substitute these values into the formula to get the sine of the target angle.

The subtraction formula complements the addition formula, providing comprehensive means to address a wide range of trigonometric problems by breaking them down into manageable parts.

Exact Trigonometric Values

Knowing exact trigonometric values is crucial for calculations involving angles like 30°, 45°, 60°, and their respective radian measures \(\pi/6\), \(\pi/4\), \(\pi/3\). These values act as building blocks for more complex trigonometric problems.

Here’s a quick guide to some key values:

• \( \sin 30^\circ = \frac{1}{2} \)
• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \cos 60^\circ = \frac{1}{2} \)

By understanding these exact values, you can effortlessly apply them to sine and cosine addition and subtraction formulas, leading to solutions involving otherwise complicated angles.

Angle Decomposition

Angle decomposition is the technique of breaking down complex angles into smaller, more manageable parts. This approach is especially beneficial when using trigonometric identities and formulas.

To effectively decompose an angle:

• Express the given angle as a sum or difference of known angles.
• Utilize the exact trigonometric values of these known angles.
• Apply addition or subtraction formulas for sine and cosine.

Through angle decomposition, we can transform an angle that initially appears difficult to handle into a series of simpler calculations. This strategy not only simplifies the problem but also helps in understanding the relationships between different trigonometric functions and angles. Making use of this concept can significantly ease the calculation of the sine or cosine of complex angles.