Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-22\) , for each given angle in standard position, determine to the nearest tenth the coordinates of the point where the terminal side intersects the unit circle. $$ 540^{\circ} $$

Step-by-Step Solution

Answer

The coordinates are \((-1, 0)\).

Step 1: Convert to a Coterminal Angle

The given angle is \(540^{\circ}\). Since it exceeds \(360^{\circ}\), we need to find a coterminal angle between \(0^{\circ}\) and \(360^{\circ}\). We can do this by subtracting \(360^{\circ}\).\[540^{\circ} - 360^{\circ} = 180^{\circ}\]So, the coterminal angle is \(180^{\circ}\).

Step 2: Determine Point on Unit Circle

The angle \(180^{\circ}\) lies along the negative x-axis of the unit circle. On a unit circle, the coordinates of the terminal side for \(180^{\circ}\) are \((-1, 0)\) because it is located exactly at one unit to the left of the origin.

Step 3: Verify the Quadrant

It's helpful to verify that we used the correct quadrant based on the angle. The angle \(180^{\circ}\) clearly lies on the negative x-axis which confirms that \((-1, 0)\) is correct and does not require further adjustments.

Key Concepts

Coterminal Angles

Coterminal angles are angles that end up in the same location on the unit circle. Even though they might have different values, they share the same terminal side. This is possible because moving in full circles on the unit circle, which is 360 degrees each time, brings you back to the same point.

• To find a coterminal angle, you can add or subtract 360 degrees from the given angle.
• For example, to find a coterminal angle for 540 degrees, since it is greater than 360 degrees, you subtract 360 and get 180 degrees as the coterminal angle.

Coterminal angles are helpful because they simplify calculations by reducing larger angles to a simpler, equivalent position on the circle.

Standard Position

Angles in standard position have a specific orientation on the coordinate plane. The key characteristic is that their vertex starts at the origin of the coordinate plane, and their initial side lies along the positive x-axis. From this point, the angle opens up in a counterclockwise direction to reach its terminal side.

• Standard position is primarily used to ensure consistency. By always starting at the positive x-axis, it provides a common point of reference for all angles.
• This alignment also aids in easily determining the angle's location relative to the x and y axes, which is crucial when working with unit circle coordinates.

Coordinates on Unit Circle

A unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is a vital tool in trigonometry and helps illustrate how angles correspond to specific points (coordinates) on the circle. Each angle has a terminal side that intersects the unit circle at a specific point.
For example, an angle of 180 degrees intersects the unit circle at the point \((-1, 0)\). This is because the angle lies directly across the origin on the negative x-axis.

• The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine of the angle.
• With these coordinates, you can easily determine trigonometric values for angles.

Understanding these intersections is essential because they are foundational in calculating sine, cosine, and even tangent values for various applications in mathematics.