Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\cos (5 \pi / 4)\) (b) \(\cos (-11 \pi / 6)\)

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{2}}{2}\); (b) \(\frac{\sqrt{3}}{2}\).

Step 1: Understand the Problem (Part A)

We need to find the exact value of \( \cos \left( \frac{5\pi}{4} \right) \). This angle is expressed in radians and lies in the third quadrant of the unit circle.

Step 2: Convert Angle (Part A)

The angle \( \frac{5\pi}{4} \) is an extension of \( \pi \) by an angle of \( \frac{\pi}{4} \) (i.e., \( \pi + \frac{\pi}{4} \)). Angles in the third quadrant have symmetry with angles in the first quadrant which helps in evaluating trigonometric functions.

Step 3: Determine Reference Angle (Part A)

The reference angle for \( \frac{5\pi}{4} \) is \( \frac{\pi}{4} \). Since this is in the third quadrant, the cosine value is negative, and we find the value for the reference angle.

Step 4: Calculate Cosine Value (Part A)

The exact cosine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \).

Step 5: Understand the Problem (Part B)

We need to find the exact value of \( \cos \left( -\frac{11\pi}{6} \right) \). This angle is negative, meaning it is measured clockwise from the positive x-axis.

Step 6: Convert Angle (Part B)

Add \( 2\pi \) to \( -\frac{11\pi}{6} \) to convert to a positive angle: \( -\frac{11\pi}{6} + 2\pi = \frac{\pi}{6} \). This means \( \cos \left( -\frac{11\pi}{6} \right) = \cos \left( \frac{\pi}{6} \right) \).

Step 7: Calculate Cosine Value (Part B)

The exact cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \). Thus, \( \cos \left( -\frac{11\pi}{6} \right) = \frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the trigonometric functions for all angles. Each point on the unit circle corresponds to an angle, measured in radians, and has coordinates

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

For example, an angle of theta corresponds to the point \(( ext{cos} \ heta, ext{sin} \ heta)\) on the circle. The circle makes it easier to visualize angles and understand their corresponding trigonometric values. The unit circle is helpful in recognizing angle cycles and symmetries, particularly between the four quadrants.

Reference Angle

A reference angle is always the acute angle formed between the terminal side of a given angle and the x-axis. It helps simplify the calculation of trigonometric functions.
Reference angles are especially useful when dealing with angles located in the second, third, or fourth quadrants, as they provide a way to relate these angles back to the first quadrant.

• In the first quadrant, the angle itself is the reference angle.
• In the second quadrant, subtract the angle from \(\pi\) (or 180 degrees) to find the reference angle.
• In the third quadrant, subtract \(\pi\) from the angle.
• In the fourth quadrant, subtract the angle from \(2\pi\) (or 360 degrees).

Reference angles simplify the process of finding trigonometric values by allowing us to use familiar angles from the first quadrant.

Cosine Value Calculation

Finding the cosine value of an angle involves determining the x-coordinate of the corresponding point on the unit circle. For many angles, these values can be memorized or calculated using the reference angle approach.
The cosine function depends on the quadrant of the angle:

• In the first quadrant, cosine values are positive, as both x and y coordinates are positive.
• In the second quadrant, cosine values become negative, as the x-coordinates are negative while y-coordinates remain positive.
• In the third quadrant, both x and y-coordinates are negative, making the cosine negative.
• In the fourth quadrant, x-coordinates turn positive again, while y-coordinates become negative.

Thus, the cosine value for any angle can be derived from its reference angle and the quadrant it is positioned in. The unit circle makes identifying these cosine values computationally easier by providing a spatial depiction of angle positions.

Quadrants in Trigonometry

The coordinate plane is divided into four quadrants, each playing a vital role in determining the signs of trigonometric functions, including cosine. Understanding these quadrants helps explain the behavior of angles and their respective trigonometric values.
The quadrants are arranged counterclockwise, beginning with the upper right:

• First Quadrant: Angles between \(0\) and \(\pi/2\) (0 to 90 degrees) have both sine and cosine values positive.
• Second Quadrant: Angles between \(\pi/2\) and \(\pi\) (90 to 180 degrees) have positive sine and negative cosine values.
• Third Quadrant: Angles between \(\pi\) and \(3\pi/2\) (180 to 270 degrees) have both sine and cosine values negative.
• Fourth Quadrant: Angles between \(3\pi/2\) and \(2\pi\) (270 to 360 degrees) have negative sine but positive cosine values.

This understanding of quadrants is significant when predicting the sign of trigonometric values for non-acute angles, making calculations accurate and coherent with the angles' spatial positioning on the unit circle.