Problem 40 - Algebra 2

Question

Draw an angle in standard position with each given measure. Then find the values of the cosine and sine of the angle to the nearest hundredth. \(-2 \pi\) radians

Step-by-Step Solution

Answer

Drawing an angle of \(-2 \pi\) radians in standard position will result in the angle pointing along the positive x-axis. The cosine and sine of this angle are 1 and 0, respectively.

Step 1: Drawing the Angle in Standard Position

Since we are given the angle measure as \(-2 \pi\) radians, this indicates a full cycle in the negative direction on the unit circle. Therefore, the terminal side of the angle will end along the positive x-axis, which is the same as its initial position.

Step 2: Determine the Cosine and Sine

The cosine of an angle is given by the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Similarly, the sine of an angle is given by the y-coordinate. However, since the angle \(-2 \pi\) radians ends where it started, along the positive x-axis, the x-coordinate (cosine) will be 1, and the y-coordinate (sine) will be 0.

Key Concepts

Angles in Standard Position

Understanding angles in standard position is a fundamental concept in trigonometry. When we say an angle is in standard position, it means the angle's vertex is placed at the origin of a coordinate plane, and its initial side lies along the positive x-axis. This is the starting point from which the angle is measured. The terminal side of the angle is where the angle stops, and the direction and length of the arc from the initial side to the terminal side define the angle's measure.

• Positive angle measures are taken counterclockwise from the initial side.
• Negative angle measures travel clockwise.

The exercise gives us an angle measured as \(-2 \pi\) radians, indicating a full negative rotation. Essentially, this takes us a full circle backward and returns us to the starting point on the positive x-axis. Understanding this setup is crucial as it helps in visualizing how angles move along the unit circle.

Unit Circle

The unit circle is a crucial tool in trigonometry because it simplifies the understanding of circular functions. It is a circle with a radius of 1 unit centered at the origin of the coordinate plane. This special circle allows for simple calculations of trigonometric values like sine and cosine.

• Each point on the unit circle corresponds to the cosine and sine of an angle.
• The coordinates of points on the circle are always in the form \((\cos\theta, \sin\theta)\).

In the exercise scenario with an angle of \(-2\pi\) radians, the end point after the full rotation lies on the positive x-axis at the point \((1, 0)\). This neat location makes calculating sine and cosine values straightforward, as you'll see next.

Cosine and Sine Values

Cosine and sine provide the x and y coordinates of points on the unit circle. The cosine of an angle tells us how far along the x-axis the terminal side of the angle is, while the sine tells us the y-axis position.
When analyzing an angle like \(-2 \pi\) radians, you can find the sine and cosine by identifying the terminal point on the unit circle. As shown in the previous sections, \(-2 \pi\) radians leaves the terminal side along the positive x-axis at the point \((1, 0)\). Therefore:

• The cosine of the angle, the x-coordinate, is \(1\).
• The sine of the angle, the y-coordinate, is \(0\).

This clarity in determining cosine and sine is one reason why the unit circle is such a valuable tool in trigonometry.