Problem 6 - Algebra 2

Question

Find the exact value of each expression. \(\cos \left(-120^{\circ}\right)\)

Step-by-Step Solution

Answer

The exact value of \(\cos(-120^{\circ})\) is \(-\frac{1}{2}\).

Step 1: Convert Angle Measure

The given angle is \(-120^{\circ}\). Angles in the unit circle can be either positive or negative. Positive angles are measured counterclockwise from the positive x-axis, while negative angles are measured clockwise. Thus, \(-120^{\circ}\) can also be represented as \(360^{\circ} + (-120^{\circ}) = 240^{\circ}\). Therefore, \(-120^{\circ}\) is equivalent to \(240^{\circ}\) in the unit circle.

Step 2: Locate the Angle in the Unit Circle

To determine \( ext{cos}(240^{\circ})\), note that \(240^{\circ}\) lies in the third quadrant of the unit circle. In this quadrant, the cosine value is negative because the x-coordinate (the cosine value) is negative.

Step 3: Determine Reference Angle

The reference angle for \(240^{\circ}\) is found by calculating how far the angle is from the nearest x-axis. So, \\(240^{\circ} - 180^{\circ} = 60^{\circ}\). The reference angle is therefore \(60^{\circ}\).

Step 4: Find Cosine of the Reference Angle

The cosine of the reference angle \(60^{\circ}\) is known to be \(rac{1}{2}\).

Step 5: Apply Sign Based on Quadrant

Since \(240^{\circ}\) is in the third quadrant and cosine is negative in the third quadrant, the cosine of \(240^{\circ}\) is \(-\frac{1}{2}\). Therefore, \( ext{cos}(-120^{\circ}) = -\frac{1}{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and helps us understand angles and trigonometric functions better. The circle is called "unit" because it has a radius of 1. It is centered on the origin of the coordinate system, making it easy to visualize and calculate trigonometric functions like sine, cosine, and tangent.
The unit circle uses radians or degrees to measure angles. In our case, we use degrees. The circle begins at 0 degrees on the positive x-axis and moves counterclockwise. An important aspect is using positive angles, which move counterclockwise, and negative angles, which move clockwise.
In a complete circle, the angle is 360 degrees or 2π radians. Navigating on this circle allows us to find specific values for trigonometric functions like cosine at any angle. For example, the angle of \(-120^{\circ}\), when converted to a positive angle on the unit circle, is \(240^{\circ}\), which lands in the third quadrant.

Reference Angle

A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis. It's always between 0 and 90 degrees. Reference angles are key to simplifying trigonometric calculations because knowing the reference angle and the quadrant helps determine the overall sign and value of a trigonometric function.
To find a reference angle, you look at how far your angle is from the nearest x-axis line. For example, if we have \(240^{\circ}\), it is in the third quadrant. We find the reference angle by subtracting from the closest x-axis, which is \(180^{\circ}\), resulting in a reference angle of \(60^{\circ}\).
Using reference angles and knowing the nature of each quadrant simplifies the process of finding the exact trigonometric values by identifying the cosine or sine in an easily manageable way:

• Quadrant I: all trigonometric functions positive
• Quadrant II: sine positive
• Quadrant III: tangent positive
• Quadrant IV: cosine positive

Cosine Function

The cosine function is one of the primary trigonometric functions and is crucial for measuring angles and determining distances on the unit circle. Cosine refers to the x-coordinate of a point on the unit circle. This means that when you find the cosine of an angle, you look at where the point intersects the unit circle on the x-axis.
Cosine values vary depending on which quadrant the angle is in, owing to the positive or negative nature of the x-coordinate in those quadrants. For our angle, which is \(240^{\circ}\) or equivalently \(-120^{\circ}\), it falls within the third quadrant. In this quadrant, cosine values are negative.
When dealing with reference angles like \(60^{\circ}\), which is the reference for \(240^{\circ}\), we know from experience that the cosine of \(60^{\circ}\) is \(\frac{1}{2}\). Because \(240^{\circ}\) resides in the third quadrant, the cosine becomes \(-\frac{1}{2}\). Therefore, the cosine of \(-120^{\circ}\) is also \(-\frac{1}{2}\), maintaining the negative characteristic due to the quadrant's influence.