Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch each angle in standard position. (a) \(-\frac{7 \pi}{4}\) (b) \(-\frac{5 \pi}{2}\)

Step-by-Step Solution

Answer

(a) The direction of rotation for \(-\frac{7 \pi}{4}\) is clockwise, and it lies in the first quadrant. (b) For \(-\frac{5 \pi}{2}\) the direction of rotation is also clockwise, and it ends up in the fourth quadrant.

Step 1: Sketching \(-\frac{7 \pi}{4}\)

Since this is a negative angle, it will be drawn clockwise from the positive x-axis. To sketch \(-\frac{7 \pi}{4}\), it can be helpful to note that \(2\pi\) is equivalent to a full circle, or \( \frac{8 \pi}{4} \). Therefore, \(-\frac{7 \pi}{4}\) is like taking one quarter turn clockwise back from a full circle.

Step 2: Sketching \(-\frac{5 \pi}{2}\)

The angle \(-\frac{5 \pi}{2}\) will be also sketched in a clockwise direction. Given that \(2\pi\) is the same as \( \frac{4 \pi}{2} \), the angle \(-\frac{5 \pi}{2}\) represents a rotation of one and a quarter full turns in the clockwise direction.

Key Concepts

Standard Position

In trigonometry, an angle is said to be in "standard position" when its vertex is placed at the origin of a coordinate plane, and its initial side lies along the positive x-axis.
This positioning is crucial for the visualization of angles, enabling consistent representation and easier computation in trigonometric functions.
Here are some additional points to consider:

• The initial side is fixed along the positive x-axis.
• Angles are measured from this initial side.
• The terminal side is the final position of the angle after rotation.

This setup allows us to easily determine the direction and magnitude of the angle, whether it is positive or negative. It forms the foundational basis for defining angle measurement in trigonometry.

Clockwise Angle Measurement

Angles measured in a clockwise direction are considered negative in trigonometry.
When dealing with such angles, it is important to understand the direction of rotation.
Here are a few key notes:

• Clockwise rotation begins from the positive x-axis.
• It is the opposite of the conventional counterclockwise direction (positive angles).
• Clockwise directions may seem backward, but they follow the same principles as positive angles.

Recognizing clockwise as the direction for negative angles helps in accurately sketching and understanding these angles, as seen in examples such as e. configurations like - 7 rac{ π 7 } 4 .

Negative Angles

Negative angles are a key concept in trigonometry and are integral to understanding directional rotation on the unit circle.
These angles are defined as those which result from a clockwise rotation, moving opposite to positive angle rotation.
Let's break it down:

• Negative angles indicate a clockwise rotation.
• The numeric value of the angle remains the same, but the direction changes.
• It's a common occurrence in coordinate geometry and trigonometry problems.

In examples such as \(-\frac{7\pi}{4}\) and \(-\frac{5\pi}{2}\), understanding negative angles simplifies the process of determining the final position of the angle on the plane.

Radians

Radians are a unit of angular measure in trigonometry, based on the radius of a circle.
This unit is often favored in mathematics because of its natural relation to the circle, making calculations more straightforward.
Consider these points:

• A radian is the angle created by taking the radius of a circle and wrapping it along the circle's edge.
• Since a circle's circumference is \(2\pi\) times the radius, a full circle contains \(2\pi\) radians.
• Radians allow direct correlation to the unit circle, simplifying the angle's conceptualization.

When working with angles such as \(-\frac{7\pi}{4}\) or \(-\frac{5\pi}{2}\), understanding radians facilitates the conversion of these measures into positions on the coordinate plane, which is crucial for sketching and trigonometric analysis.