Question

Without using a calculator, evaluate, if possible, the following expressions. $$\cos ^{-1}\left(-\frac{1}{2}\right)$$

Short answer

Answer: The angle whose cosine value is equal to -1/2 is 120 degrees or (2π/3) radians.

Step-by-step solution

Identify the cosine value

We are given an expression to evaluate without using a calculator: $$\cos ^{-1}\left(-\frac{1}{2}\right)$$ This expression is asking for the angle whose cosine value is equal to -1/2.

Recall the unit circle and trigonometric functions

The cosine function represents the x-coordinate of a point on the unit circle that corresponds to a given angle. In order to find the angle that corresponds to a cosine value of -1/2, we need to think about the unit circle.

Identify the angle in the unit circle

To find the angle with a cosine value of -1/2, we must look at the unit circle and focus on the angles whose x-coordinate is -1/2. We know that cosine is negative in the second and third quadrants. The angles that result in a cosine value of 1/2 are 60 degrees (π/3 radians) and 120 degrees (2π/3 radians). Therefore, the angle with a cosine value of -1/2 must be in the second quadrant and be the supplement of 60 degrees (π/3 radians).

Calculate the angle

Since the angle is supplementary to 60 degrees, we subtract 60 degrees from 180 degrees (π radians): $$180^{\circ} - 60^{\circ} = 120^{\circ}$$ Or, in radians: $$\pi - \frac{\pi}{3} = \frac{2\pi}{3}$$

Write the final answer

The angle whose cosine value is -1/2 is: $$\cos ^{-1}\left(-\frac{1}{2}\right) = 120^{\circ}= \frac{2\pi}{3}$$

Key concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle helps us understand how trigonometric functions like sine and cosine relate to angles.

For any angle, the coordinates of the corresponding point on the unit circle reveal valuable information:

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

Since the circle has a radius of 1, every point lies exactly one unit away from the origin. This makes calculating cosine and sine values straightforward.
The unit circle also shows how these functions repeat in a periodic manner. As you travel around the circle, the values of sine and cosine repeat every 360 degrees or \(2\pi\) radians.

Cosine Function

The cosine function is key to understanding the unit circle. It measures the horizontal distance of a point from the origin along the x-axis. For any given angle \(\theta\), the cosine value is the x-coordinate of the corresponding point on the unit circle.

Here's how you can visualize it:

• When the angle is 0 degrees or 0 radians, the value of cosine is 1.
• At 90 degrees, or \(\frac{\pi}{2}\) radians, the cosine value drops to 0.
• For 180 degrees or \(\pi\) radians, cosine becomes -1.

Cosine values can range between -1 and 1, making it periodic with peaks and troughs. These x-coordinates shift between positive and negative values depending on the angle's quadrant, reflecting its periodic oscillation.

Trigonometric Quadrants

Trigonometric quadrants divide the coordinate plane into four sections. Each section represents a different range of angles and determines the sign of sine, cosine, and tangent values. Understanding these quadrants is crucial when analyzing the cosine function and other trigonometric functions.

Here's a breakdown of the quadrants:

• First Quadrant: 0 to 90 degrees. All trigonometric values (sine, cosine, and tangent) are positive here.
• Second Quadrant: 90 to 180 degrees. Sine values are positive, but cosine and tangent are negative.
• Third Quadrant: 180 to 270 degrees. Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant: 270 to 360 degrees. Cosine is positive, and sine and tangent are negative.

In the given exercise, the expression \(\cos^{-1}(-\frac{1}{2})\) leads us to focus on the second quadrant. This is because cosine is negative there, helping us determine the correct angle associated with this cosine value.