Problem 14 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \cos \left(-60^{\circ}\right) $$

Step-by-Step Solution

Answer

\( \cos(-60^{\circ}) = \frac{1}{2} \).

Step 1: Identify Trigonometric Identity

We start by using the trigonometric identity for cosine of negative angles. Using the identity: \( \cos(-\theta) = \cos(\theta) \), we can simplify our problem to \( \cos(-60^{\circ}) = \cos(60^{\circ}) \).

Step 2: Recall Standard Value

Recall that \( \cos(60^{\circ}) \) is a standard angle in trigonometry. From the unit circle or the known values, \( \cos(60^{\circ}) \) is equal to \( \frac{1}{2} \).

Step 3: Verify the Result Using the Unit Circle

To verify, imagine the position of 60 degrees on the unit circle. It is located in the first quadrant where cosine values are positive, confirming that \( \cos(60^{\circ}) = \frac{1}{2} \).

Key Concepts

Cosine of Angles

The trigonometric function cosine, often abbreviated as cos, is a fundamental function used in many areas of mathematics. For any angle \( \theta \), the cosine of that angle represents the x-coordinate of a point on the unit circle.

The cosine value can be determined for any angle, whether it is in degrees or radians.

• For angles in the first quadrant (0° to 90°), cosine values are positive.
• For angles in the second quadrant (90° to 180°), cosine values are negative.
• In the third quadrant (180° to 270°), cosine values remain negative.
• For the fourth quadrant (270° to 360°), cosine values turn positive again.

A standard angle, like 60°, is one that has a widely recognized cosine value. Such angles are easy to recall, facilitating quick calculations.

Negative Angle Identity

The negative angle identity for cosine makes it easier to work with angles measuring less than zero. The equation is:
\[ \cos(-\theta) = \cos(\theta) \]This simplification indicates that negative angles produce the same cosine value as their positive counterparts.

This is because cosine is an even function, meaning its graph is symmetric about the y-axis. Therefore, whether you rotate clockwise or counterclockwise, the horizontal distance (or x-coordinate) from the origin will be identical. This property is useful for solving problems that involve negative angles, as it simplifies calculations and reduces computational demands by converting it to a positive angle equivalent.

Unit Circle

The unit circle is a critical concept in trigonometry, serving as a visual aid for understanding trigonometric functions. The circle has a radius of one unit, centered at the origin (0,0) of the coordinate plane.

Each point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis.

• The coordinates of any point \( (x, y) \) on the unit circle are standardized as \( (\cos(\theta), \sin(\theta)) \).
• This means for an angle of 60°, the point will be \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \), confirming \( \cos(60^{\circ}) = \frac{1}{2} \).

The unit circle assists in quickly identifying trigonometric values, especially with angles such as 30°, 60°, and 90°, due to their symmetrical properties.

Utilizing the unit circle can enhance understanding and facilitate mastery over trigonometric functions and their corresponding values.