Problem 35 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \cos \left(-300^{\circ}\right) $$

Step-by-Step Solution

Answer

The value of \(\cos(-300^{\circ})\) is 0.5

Step 1: Find the equivalent angle in the positive direction

We have a negative angle -300 degrees here. We know that cosine has a periodicity of 360 degrees. This means we can add 360 degrees to our angle until we get a positive angle without changing the value of the cosine function. Hence, we get: \(-300^{\circ} + 360^{\circ} = 60^{\circ}\)

Step 2: Use the known trigonometric value

By memorizing the unit circle or from the 30-60-90 degree triangle, we know the value of cosine of 60 degrees.Hence, we can say: \(\cos(60^{\circ}) = 0.5\)

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions, such as sine and cosine, repeat their values over regular intervals. This repeat pattern is known as periodicity. For the cosine function, this means that if you add or subtract 360 degrees (or \(2\pi\) radians), the cosine value remains unchanged.

• The period of the cosine function is 360 degrees or \(2\pi\) radians.
• Regardless of the angle measure, adding or subtracting full periods (like 360 degrees) will not affect the value of the function.

In our exercise, we were given an angle of -300 degrees. By adding 360 degrees, we found its positive equivalent, 60 degrees.
This demonstrates the periodicity concept:

• Any negative angle can be converted into a positive angle, resulting in the same cosine value.
• This conversion is very useful, especially when identifying angles on the unit circle.

Cosine Function

The cosine function is one of the primary trigonometric functions used to relate angles to the sides of a triangle, particularly in relation to the unit circle. Cosine measures the x-coordinate of a point on the unit circle corresponding to a given angle.

• The cosine function is denoted as \(\cos(\theta)\), where \(\theta\) represents the angle.
• It ranges between -1 and 1, depending on the angle \(\theta\).

For 60 degrees, the cosine value is exactly 0.5. It is derived from either the unit circle or the properties of the 30-60-90 triangles.
Some important characteristics of the cosine function include:

• It is an even function: \(\cos(-\theta) = \cos(\theta)\).
• It has zeros at odd multiples of 90 degrees (e.g., 90 degrees, 270 degrees).
• It reaches its maximum value of 1 at 0 and 360 degrees and its minimum of -1 at 180 degrees.

Understanding cosine's characteristics helps when dealing with trigonometric identities and solving trigonometric equations.

Unit Circle

The unit circle is a fundamental concept in trigonometry, providing a simple way to understand angles and trigonometric values. A unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle's cosine and sine values.

• The x-coordinate of each point on the unit circle represents the cosine value of the angle.
• The y-coordinate gives the sine value.

For example, at 60 degrees (or \(\pi/3\) radians), the point on the unit circle has coordinates \((1/2, \sqrt{3}/2)\). Here, the x-coordinate, 1/2, corresponds to the cosine value of 60 degrees.
Visually, the unit circle helps students remember and derive trigonometric identities and values.

• Key angles, like 0, 30, 45, 60, and 90 degrees, are often memorized in terms of their unit circle coordinates.
• The unit circle assists in understanding the periodic nature of trigonometric functions.