Problem 57 - Precalculus

Question

Evaluate the following expressions exactly: $$\cos 240^{\circ}$$

Step-by-Step Solution

Answer

\( \cos 240^{\circ} = -\frac{1}{2} \).

Step 1: Identify the Reference Angle

The given angle is \( 240^{\circ} \). This angle lies in the third quadrant of the unit circle. To find the reference angle, we subtract \( 180^{\circ} \) from \( 240^{\circ} \): \( 240^{\circ} - 180^{\circ} = 60^{\circ} \). Therefore, the reference angle is \( 60^{\circ} \).

Step 2: Determine the Cosine of the Reference Angle

The cosine of the reference angle \( 60^{\circ} \) is \( \cos 60^{\circ} = \frac{1}{2} \).

Step 3: Adjust for the Quadrant

In the third quadrant, the cosine function is negative. Therefore, we must take the negative of the cosine of the reference angle. Thus, \( \cos 240^{\circ} = -\frac{1}{2} \).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of 1 unit centered at the origin of a coordinate system. This makes it particularly useful in defining trigonometric functions such as sine and cosine. The circle allows us to easily visualize angles and their corresponding trigonometric values.

Here's why it's special:

• Every angle corresponds to a point on the unit circle.
• The x-coordinate of a point on the unit circle equals the cosine of the angle.
• The y-coordinate equals the sine of the angle.
• Angles can be measured in degrees or radians as they wrap around the circle.

Understanding the unit circle helps you get a grasp on how angles and their trigonometric functions interact. In our exercise, knowing that 240° lies in the third quadrant is crucial because it affects the sign of the cosine we are looking for.

Reference Angle

A reference angle is the smallest angle between the terminal side of an angle and the horizontal axis. It helps in simplifying the process of finding the trigonometric values of any angle, especially those outside the 0° to 90° range.

Here’s how to determine it:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from 180°.
• In the third quadrant, subtract 180° from the angle.
• For the fourth quadrant, subtract the angle from 360°.

In our example, the angle given was 240°. It is located in the third quadrant. To find its reference angle, you subtract 180° from 240°, resulting in a reference angle of 60°. Understanding the reference angle allows us to use standard trigonometric values to find the functions like cosine, by simply adjusting the sign according to the quadrant.

Cosine Function

The cosine function is one of the primary trigonometric functions and is crucial in various applications including geometry, physics, and engineering. For a given angle θ, cosine gives the x-coordinate of the corresponding point on the unit circle.

Key points about the cosine function:

• In the unit circle, cosine values range from -1 to 1.
• The cosine of an angle tells us how much of the angle's rotation corresponds to movement along the horizontal axis.
• It's periodic with a period of 360° or 2π radians, meaning the values repeat after this interval.

When calculating the cosine of larger angles like 240°, the quadrant determines the sign. The third quadrant, where our angle is located, gives a negative cosine. Therefore, even though the reference angle 60° yields a cosine of 1/2, the actual cosine of 240° is negative, specifically -1/2. This is why understanding quadrants and their effect is essential for correctly evaluating trigonometric expressions.