Problem 329 - Precalculus

Question

For the following exercises, draw the angle provided in standard position on the Cartesian plane. Find the exact value of \(\cos \pi\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Angle

The angle provided is \( \pi \) radians. In standard position, this angle is measured from the positive \( x \)-axis counterclockwise. One complete revolution around the circle is \( 2\pi \) radians, so \( \pi \) radians represents a half-turn.

Step 2: Identifying the Terminal Side

Since \( \pi \) radians is half of \( 2\pi \) radians, the terminal side of the angle will lie exactly on the negative \( x \)-axis. In standard position, start from the positive \( x \)-axis and move counterclockwise \( \pi \) radians to reach the negative \( x \)-axis.

Step 3: Finding the Cosine Value

The cosine of an angle in the unit circle is the \( x \)-coordinate of the point where the terminal side of the angle intersects the unit circle. For the angle \( \pi \), this point is \((-1, 0)\) on the unit circle.

Step 4: Determine the Exact Value

Since the point \((-1, 0)\) lies on the unit circle, and the \( x \)-coordinate is \(-1\), the exact value of \( \cos(\pi) \) is \(-1\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand how angles relate to coordinates on a plane. A unit circle has a radius of 1 unit and is centered at the origin of a Cartesian coordinate system. Because the radius is 1, any point on the circle can be defined using the cosine and sine functions.

In the unit circle, an angle is formed by a line from the origin to a point on the circle, measured from the positive x-axis. This circle is essential for defining trigonometric functions, as it relates angles to coordinate points.

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

This relationship makes it easy to find trigonometric values for any angle using the unit circle.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the circle’s circumference. One complete revolution around the circle is equivalent to some 2𝜋 radians.

To understand radians, remember that the length of the arc is equal to the radius times the angle in radians. This makes radians a natural unit for measuring angles, especially in the context of the unit circle.

• One radian is the angle formed when the arc length is equal to the radius.
• Thus, 2𝜋 radians correspond to 360 degrees, making 𝜋 radians equivalent to 180 degrees.

Using radians simplifies working with angles in trigonometry, as they more directly relate angles to the geometry of the circle.

Cosine Function

The cosine function is a fundamental trigonometric function that relates an angle in a circle to its x-coordinate on the unit circle. For any angle, the cosine value is determined by where the terminal side of the angle intersects the unit circle.

For our specific example, we are dealing with the angle \( \pi \) radians. In this case:

• The terminal point for the angle \( \pi \) lies on the negative x-axis and corresponds to the point (-1, 0) on the unit circle.
• Therefore, the cosine of \( \pi \) is just the x-coordinate of this point.
• This results in \( \cos(\pi) = -1 \).

The cosine function is periodic, meaning it repeats its values in a regular pattern: every 2𝜋 radians or 360 degrees. This is particularly useful for analyzing and synthesizing waves and oscillations in various fields such as physics and engineering.

Standard Position

An angle in standard position is an important concept for understanding angles in the Cartesian plane. By convention, an angle's vertex is placed at the origin, and one ray, known as the initial side, lies along the positive x-axis.

When measuring an angle in standard position:

• The angle is measured counterclockwise from the initial side.
• If the angle is negative, it is measured clockwise.
• The terminal side is the other ray that defines the angle, starting from the origin.

For instance, the angle \( \pi \) radians mentioned in our problem is placed in standard position. Its terminal side is on the negative x-axis, which aids in finding the angle's trigonometric functions using the unit circle. This method of placing angles makes it easier to visualize and solve problems related to trigonometry.