Problem 14 - Precalculus Mathematics for Calculus

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: Recognize the Property of Cosine Function

The cosine function is even, which means that for any angle \( \theta \), \( \cos(-\theta) = \cos(\theta) \). Therefore, we have \( \cos(-60^{\circ}) = \cos(60^{\circ}) \).

Step 2: Determine the Cosine of 60 Degrees

Using the unit circle or common trigonometric values, \( \cos(60^{\circ}) \) is known to be \( \frac{1}{2} \).

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions that involves the relationship between an angle in a right triangle and the ratio of the adjacent side length to the hypotenuse. In the context of the unit circle, the cosine function is defined as the x-coordinate of an angle plotted in the coordinate plane. As the angle increases, the cosine value changes and forms a wave-like pattern, known as the cosine wave.

Cosine values can range from -1 to 1. These values repeat every 360° (or 2π radians) due to the periodic nature of trigonometric functions. This concept is crucial in many fields, such as engineering and physics, because it helps describe oscillations, waves, and rotational dynamics.

Unit Circle

The unit circle is a powerful tool for understanding and calculating trigonometric functions. It is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. In this circle, the angle's cosine value corresponds to the x-coordinate of the point on the circle’s circumference, while the sine value corresponds to the y-coordinate.

By using the unit circle, we can find cosine, sine, and tangent values for various angles. When you plot an angle from the positive x-axis, you're effectively measuring how far around the unit circle the angle extends, making it easier to visualize and compute trigonometric values.

Even Functions

An even function has a special symmetry in its graph: it is mirror-symmetrical about the y-axis. This means that for any even function, the function's value at a negative input ( \(-x\)) is the same as the value at the corresponding positive input ( \(x\)). For the cosine function, this property is denoted as \(\cos(-\theta) = \cos(\theta)\).

This property simplifies calculations and helps to predict values for negative angles without needing to use additional tools. For instance, \(\cos(-60^\circ)\) is the same as \(\cos(60^\circ)\), which is a significant time-saver when working through trigonometric problems.

Common Angles

In trigonometry, certain angles have widely known sine, cosine, and tangent values, referred to as common angles. These include 0°, 30°, 45°, 60°, and 90° (as well as their radian equivalents: 0, \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\)). These angles can often be found on the unit circle, or they can be memorized because of their frequent use.

For instance, knowing that \(\cos(60^\circ) = \frac{1}{2}\) allows quick computation in trigonometric problems. These angles serve as a cornerstone for more complex calculations in trigonometry and are essential for students aiming to understand deeper mathematical concepts.