Problem 93 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \cos \pi $$

Step-by-Step Solution

Answer

The exact value of \( \cos \pi \) is -1.

Step 1: Understanding the Angle

We need to find the value of the cosine function at \( \pi \) radians. Recall, \( \pi \) radians corresponds to 180 degrees in the unit circle, which is a half-circle movement counterclockwise starting from the positive x-axis.

Step 2: Locate the Angle on the Unit Circle

On the unit circle, \( \pi \) (or 180 degrees) is located at the negative x-axis (leftmost point). This corresponds to the coordinate (-1, 0).

Step 3: Determine the Cosine Value

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. Therefore, for the angle \( \pi \), the cosine value is -1.

Key Concepts

cosine function

The cosine function is a fundamental concept in trigonometry often denoted as "cos". This function is used to determine the x-coordinate of a point on the unit circle that corresponds to a given angle. In simpler terms, it tells us how far left or right the point is from the origin, based on its position around the circle.

• The cosine function is an even function, meaning that cos(-θ) = cos(θ) for any angle θ.
• The values of the cosine function range from -1 to 1.
• At 0 radians (or 0 degrees), the cosine value is 1, as the point is on the positive x-axis.

These properties make the cosine function crucial for understanding the relationships between angles and coordinates on the unit circle.

unit circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is a principal tool in trigonometry, especially for understanding the properties of trigonometric functions like sine and cosine. Each point on the unit circle corresponds to an angle and can be described using the coordinates (cos(θ), sin(θ)), where θ is the angle in radians.

• The circle's circumference can be used to explain the periodic nature of trigonometric functions.
• Any point on the unit circle can be represented by the coordinate (x, y), where x = cos(θ) and y = sin(θ).
• For example, at 0 radians, the point is (1, 0); at π/2 radians, it is (0, 1); and at π radians, it is (-1, 0).

By using the unit circle, we can visualize not only the position of angles but also the corresponding cosine and sine values, which highlight their cyclical nature.

angle measurement

Angle measurement in trigonometry can be done in different units, typically degrees and radians. Understanding these units is essential for working with the unit circle and trigonometric functions.

• Degrees measure angles based on dividing a circle into 360 parts. So, one complete circle is 360 degrees.
• Radians provide a more mathematical approach, where one complete circle is 2π radians.
• The conversion between degrees and radians is vital: π radians equals 180 degrees.

When dealing with the unit circle, radians are often used since they relate more directly to the circle’s equation and are integral in calculus. For example, the angle of π radians refers to a point on the circle directly opposite to the starting point on the positive x-axis, which has significant implications for solving trigonometric functions like cos(π).