Problem 167 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \sin \left(\frac{-9 \pi}{4}\right) \cos \left(\frac{-\pi}{6}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Angles

The given angles are \( \frac{-9 \pi}{4} \) and \( \frac{-\pi}{6} \). These angles need to be evaluated using trigonometric functions on a graphing calculator.

Step 2: Convert to Positive Angles

Since angles can be represented with equivalent positive angles, we find equivalent angles for easier evaluation. \( \frac{-9\pi}{4} \) is the same as \( \frac{-9\pi}{4} + 2\pi \times 3 = \frac{3\pi}{4} \) (adding \( 6\pi/4 \) ), and \( \frac{-\pi}{6} \) is the same as \( \frac{-\pi}{6} + 2\pi \times 1 = \frac{11\pi}{6} \).

Step 3: Calculate Each Trigonometric Function

Using the graphing calculator, evaluate each trigonometric function:\( \sin \left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \)\( \cos \left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \)

Step 4: Multiply the Results

Multiply the results of the trigonometric functions:\( \sin \left(\frac{3\pi}{4}\right) \times \cos \left(\frac{11\pi}{6}\right) = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4} \).

Key Concepts

Angle Conversion

In trigonometry, it is common to work with negative angles or angles larger than 360 degrees (or 2π radians). Converting these into positive angles that fall within one complete circle—between 0 and 2π—will often simplify calculations.
To convert the given angles, like \( \frac{-9\pi}{4} \) and \( \frac{-\pi}{6} \), we can add \( 2\pi \) repeatedly until the angle is positive:

• \( \frac{-9\pi}{4} \) becomes \( \frac{3\pi}{4} \) after adding \( 2\pi \times 3 \)
• \( \frac{-\pi}{6} \) becomes \( \frac{11\pi}{6} \) after adding \( 2\pi \times 1 \)

By doing this, you obtain equivalent angles, making it easier to evaluate trigonometric functions.

Graphing Calculator

Using a graphing calculator is an essential skill in evaluating trigonometric functions, especially when dealing with complex angles. A graphing calculator not only aids in simplifying the process of angle conversion and evaluation but also visualizes these angles in terms of their sine and cosine values.
Here’s how a graphing calculator can assist you:

• Input your positive angle values: like \( \frac{3\pi}{4} \) and \( \frac{11\pi}{6} \).
• Directly obtain the sine and cosine values without manual computations.
• Double check calculations, especially when dealing with fractions and radicals.

Graphing calculators save time, minimize errors, and build understanding of trigonometric concepts—an invaluable tool in math and science applications.

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry as they describe the relationships between angles and side lengths in right triangles. For example, in the unit circle, sine represents the vertical line segment relating to the angle, while cosine represents the horizontal segment.
When working through an equation like \( \sin(\frac{3\pi}{4}) \), it helps to remember:

• \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \) - derived from its position in the unit circle in the 2nd quadrant.
• \( \cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2} \) - identified from its position near the positive x-axis.

These functions are foundational for solving more complex trigonometric identities and problems.

Angle Equivalence

Angle equivalence is about understanding that rotating in either clockwise or counterclockwise direction leads to angles that are equivalent in their trigonometric characteristics. The key is recognizing that a complete rotation adds \( 2\pi \) (or 360 degrees).
This principle lets us change -\( \frac{9\pi}{4} \) and other negative angles into their positive, easier-to-work-with equivalents, but without altering their trigonometric values. For instance:

• A negative angle like \( \frac{-9\pi}{4} \) is equated to its positive counterpart \( \frac{3\pi}{4} \).

This technique helps to access standardized values for sine and cosine, aiding both in manual and calculator-based evaluations for solving equations or graphing.