Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \cos 420^{\circ} $$

Short answer

The exact value of \(\cos 420^{\circ} \) is \(\frac{1}{2}\).

Step-by-step solution

Understand Trigonometric Function Periodicity

The cosine function is periodic with a period of 360°. This means that \(\forall n \in \mathbb{Z}, \cos(\theta + 360^{\circ} \cdot n) = \cos(\theta)\).

Subtract Multiples of 360°

Subtract 360° from 420° to find an equivalent angle within the first period: \[ 420^{\circ} - 360^{\circ} = 60^{\circ} \]

Find the Exact Value of \(\cos 60^{\circ} \)

Using the unit circle or trigonometric table, the exact value of \(\cos 60^{\circ} \) is \(\frac{1}{2}\).

Key concepts

cosine function

The cosine function is one of the main trigonometric functions in mathematics. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function is usually written as \(\text{cos}(\theta)\). This function is very important in various fields such as engineering, physics, and even in everyday calculations concerning waves and oscillations.
The cosine function has several important properties:

• It ranges between -1 and 1, inclusive.
• It is an even function, which implies that \(\text{cos}(\theta) = \text{cos}(-\theta)\).
• It has a period of 360˚ for degrees and \(2\text{π}\) for radians, meaning that \(\text{cos}(\theta + 360^{\text{˚}}) = \text{cos}(\theta)\).
• It starts from 1 when \(\theta\) is 0.

Understanding these properties can help in resolving many trigonometric problems easily.

periodic functions

Periodicity in trigonometric functions refers to the repeating nature of their values over specific intervals. For the cosine function, this interval—or period—is 360˚ in degrees or \(2\text{π}\) in radians. The periodic behavior means that for any angle \(\theta\), its cosine value will repeat after every 360˚. This property can simplify many trigonometric problems. For instance, in the given exercise, we have to find \(\text{cos}(420^\text{˚})\). Instead of directly evaluating it, we can use its periodic nature.
By subtracting multiples of its period (360˚), we adjust \(\theta\) to fall within the basic range of 0˚ to 360˚:

• \(420^\text{˚} - 360^\text{˚} = 60^\text{˚}\)

This makes the task much simpler and allows us to find the cosine value easily.

exact values in trigonometry

Certain angles in trigonometry have well-known exact values for cosine, sine, and other trigonometric functions. These angles include 0˚, 30˚, 45˚, 60˚, and 90˚. Memorizing these values helps greatly in solving trigonometric problems quickly. For example, you should remember the exact value of \(\text{cos}(60^\text{˚})\) which is \(\frac{1}{2}\).
Exact values can also be found using the unit circle. Here are some exact values to remember for cosine:

• \(\text{cos}(0^\text{˚}) = 1\)
• \(\text{cos}(30^\text{˚}) = \frac{\sqrt{3}}{2}\)
• \(\text{cos}(45^\text{˚}) = \frac{\sqrt{2}}{2}\)
• \(\text{cos}(60^\text{˚}) = \frac{1}{2}\)
• \(\text{cos}(90^\text{˚}) = 0\)

Having these values at your fingertips allows for quick and accurate calculations on exams and in real-world applications.

unit circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate system (0,0). This circle helps visualize and define the trigonometric functions for all angles.
On the unit circle, any angle \(\theta\) corresponds to a point \((x, y)\) on the circle. For that point:

• \(x = \text{cos}(\theta)\)
• \(y = \text{sin}(\theta)\)

Thus, the unit circle provides a geometric way to find trigonometric values. For example, when you need to find \(\text{cos}(60^\text{˚})\), you look at the point on the unit circle that aligns with 60˚. From this, you derive that the cosine value is \(\frac{1}{2}\). Using the unit circle makes understanding and calculating trigonometric functions more intuitive and visual.