Problem 58 - Precalculus

Question

Evaluate the following expressions exactly: $$\cos 120^{\circ}$$

Step-by-Step Solution

Answer

The exact value is \(-\frac{1}{2}\).

Step 1: Understand the Cosine Angle

We need to evaluate \( \cos 120^{\circ} \). Notice that \(120^{\circ}\) is an angle in the second quadrant of the unit circle.

Step 2: Find Reference Angle

The reference angle for \(120^{\circ}\) is found by subtracting from \(180^{\circ}\) (since \(120^{\circ}\) is in the second quadrant). Thus, the reference angle is \( 180^{\circ} - 120^{\circ} = 60^{\circ} \).

Step 3: Use the Cosine Function

In the second quadrant, the cosine of an angle is negative, and the cosine of the reference angle \(60^{\circ}\) is \(\frac{1}{2}\). Therefore, \( \cos 120^{\circ} = -\frac{1}{2} \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that reflects the horizontal coordinate of an angle on the unit circle. It measures how far along the x-axis the point corresponding to an angle is, as you move counterclockwise from the positive x-axis.

• The cosine of an angle in a right triangle is the ratio of the adjacent side over the hypotenuse.
• The cosine function is periodic with a cycle of 360° (or 2π radians), meaning it repeats every full rotation of the circle.
• In the unit circle, as the angle increases from 0° to 360°, the cosine value starts at 1, decreases to -1, and returns to 1.

When working with angles beyond the first quadrant (0° to 90°), it's important to remember that the cosine value can be both positive and negative depending on the quadrant:

• First Quadrant (0° to 90°): Cosine values are positive.
• Second Quadrant (90° to 180°): Cosine values are negative.
• Third Quadrant (180° to 270°): Cosine values remain negative.
• Fourth Quadrant (270° to 360°): Cosine values return to positive.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a powerful tool in trigonometry to understand the relationship between cosine and other trigonometric functions. Here's why the unit circle is essential:

• Standard Reference: The unit circle provides a standard way to express cosine (and sine) values for different angles.
• Quadrants and Signs: As a circle is divided into four quadrants, it helps determine whether the trigonometric functions are positive or negative.
• Angles in Radians and Degrees: The unit circle allows conversion between angle measures: degrees and radians. For example, 360° corresponds to 2π radians.

By positioning the terminal side of angles along the circumference of the unit circle, you can easily calculate and memorize cosine values for angles:

• 0° and 360° correspond to a cosine value of 1.
• 180° gives a cosine of -1.
• 90° and 270° yield a cosine of 0.

Reference Angle

A reference angle is a tool to simplify finding trigonometric function values for any given angle. It is the smallest angle a given angle makes with the x-axis. For angles greater than 90°, reference angles keep calculations manageable:

• For the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from 180°. For example, for 120°, the calculation is: 180° - 120° = 60°.
• For the third quadrant, subtract 180° from the angle.
• In the fourth quadrant, subtract the angle from 360°.

Understanding the concept of reference angles makes it easier to find the cosine (and other trigonometric function values) for non-standard angles:

• Reference angles allow you to use a known base angle to find trigonometric values in another quadrant.
• This is particularly useful in conjunction with the signs determined by the unit circle quadrants for accurate calculations.