Problem 27 - Precalculus Mathematics for Calculus

Question

Use an appropriate Half-Angle Formula to find the exact value of the expression. $$\sin \frac{9 \pi}{8}$$

Step-by-Step Solution

Answer

The exact value of \( \sin \frac{9\pi}{8} \) is \( -\frac{\sqrt{2 - \sqrt{2}}}{2} \).

Step 1: Identify the Half-Angle Formula for Sine

The half-angle formula for sine is \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \). To determine the correct sign, we need to consider the quadrant in which \( \frac{9\pi}{8} \) lies. Since \( \frac{9\pi}{8} \) is slightly more than \( \pi \) (which is equivalent to \( 180^{\circ} \)), it lies in the third quadrant where sine is negative.

Step 2: Express Given Angle in Terms of Double the Angle

The half-angle formula requires us to express \( \frac{9\pi}{8} \) as \( \frac{\theta}{2} \), meaning \( \theta = 2 \times \frac{9\pi}{8} = \frac{9\pi}{4} \). Let's simplify \( \frac{9\pi}{4} \) into a more familiar angle.

Step 3: Simplify \( \theta \) to a Known Angle

Since \( 2\pi = \frac{8\pi}{4} \), we can subtract from \( \frac{9\pi}{4} \) to find \( \frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4} \). Therefore, \( \cos\theta \) is based on \( \theta = \frac{\pi}{4} \).

Step 4: Use the Half-Angle Formula

Now, compute \( \cos \theta = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Substitute this into the half-angle formula: \( \sin \frac{9\pi}{8} = - \sqrt{\frac{1 - \cos \frac{\pi}{4}}{2}} = - \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \).

Step 5: Simplify the Expression

To simplify \( - \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \), start with the numerator: \( 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \). Now plug it back: \( \frac{1 - \frac{\sqrt{2}}{2}}{2} = \frac{2 - \sqrt{2}}{4} \). Therefore, \( \sin \frac{9\pi}{8} = - \sqrt{\frac{2 - \sqrt{2}}{4}} = - \frac{\sqrt{2 - \sqrt{2}}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that are always true for the relationships between various trigonometric functions.
These identities are crucial for simplifying and solving complex trigonometric equations. Let's look at some fundamental identities that are often used in trigonometry:

• **Pythagorean Identities**: These include equations like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• **Reciprocal Identities**: These define relationships such as \( \sin \theta = \frac{1}{\csc \theta} \).
• **Half-Angle Identities**: These provide a way to calculate the trigonometric functions of half angles, such as \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \).

Particularly, the half-angle identities help in finding trigonometric values for angles that are not typically found on the unit circle.
In our exercise, the half-angle formula is essential for calculating the sine of \( \frac{9\pi}{8} \). By using these identities, you can convert complex trigonometric expressions into simpler forms, making the calculations more manageable.

Exact Trigonometric Values

Exact trigonometric values are specific numerical values for common angles that we often use in trigonometry.
These values are typically derived from special triangles like the 45°-45°-90° and 30°-60°-90° triangles and are important for solving trigonometric equations without a calculator.

• For example, the sine and cosine of 45° or \( \frac{\pi}{4} \) are both \( \frac{\sqrt{2}}{2} \), as used in our solution above.
• The values for 30° (or \( \frac{\pi}{6} \)) and 60° (or \( \frac{\pi}{3} \)) include \( \sin \frac{\pi}{6} = \frac{1}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \).

Understanding these exact values allows you to use trigonometric identities effectively and simplify expressions like the half-angle formula.
In this exercise, we leveraged the exact trigonometric value of \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) to calculate \( \sin \frac{9\pi}{8} \) precisely without approximation.

Quadrants of the Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand the sign and value of trigonometric functions at different angles.
It is a circle with a radius of 1 centered at the origin of a coordinate plane. The circle is divided into four quadrants, and each quadrant determines the sign of sine, cosine, and tangent values:

• **First Quadrant (0 to \( \frac{\pi}{2} \))**: Here, both sine and cosine values are positive.
• **Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \))**: The sine is positive, while the cosine is negative.
• **Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \))**: Both sine and cosine are negative. This is where \( \frac{9\pi}{8} \) falls, resulting in a negative sine value.
• **Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \))**: The sine is negative, and the cosine is positive.

Knowing which quadrant an angle lies in helps determine the appropriate signs when using identities like the half-angle formula.
It was crucial for finding the correct sign of \( \sin \frac{9\pi}{8} \), as the angle was located in the third quadrant.