Problem 14 - Precalculus

Question

For the following exercises, draw an angle in standard position with the given measure. $$ \frac{\pi}{2} $$

Step-by-Step Solution

Answer

Draw a 90-degree counter-clockwise angle from the positive x-axis to the positive y-axis.

Step 1: Understand the Angle Measure

The angle measure given is \( \frac{\pi}{2} \) radians. In terms of degrees, this is equivalent to 90 degrees since \( \pi \) radians equals 180 degrees. This means \( \frac{\pi}{2} \) is a right angle.

Step 2: Plot the Initial Side

An angle in standard position starts with the initial side on the positive x-axis. So, start by drawing a ray on the positive x-axis of your coordinate system. This is the initial position of the angle.

Step 3: Determine the Direction of Rotation

Since the angle is positive and measured in radians, the rotation will be counter-clockwise from the initial side.

Step 4: Draw the Final Angle Position

From the initial side on the positive x-axis, rotate counter-clockwise by 90 degrees or \( \frac{\pi}{2} \) radians to get the terminal side of the angle. The terminal side will lie on the positive y-axis.

Step 5: Label the Angle

Finally, label the angle as being \( \frac{\pi}{2} \) radians. Ensure it is clear that the angle drawn is from the positive x-axis to the positive y-axis, indicating a right angle.

Key Concepts

Angle Measure in Radians

When we talk about angle measurement, we often use radians. Radians are a unit that allow us to measure angles using the radius of a circle.
Remember, one full revolution around the circle measures 2π radians, which is equivalent to 360 degrees. That means:

• π radians = 180 degrees
• ½π (or π/2) radians = 90 degrees

So, when you’re dealing with radians, you’re essentially working with fractions of π which helps in precise calculations, especially in trigonometry. Using radians simplifies many mathematical equations by linking angles directly to the radius and circumference of the circle.

Right Angle

A right angle is a fundamental concept in geometry. It equals 90 degrees or π/2 radians.
This angle usually forms an "L" shape and is the angle of each corner in a square or rectangle. In a coordinate system, a right angle changes direction entirely from one axis to another, for example:

• The terminal side of a π/2 radians angle moves from the positive x-axis to the positive y-axis.

It's essential in defining perpendicular lines and right-angled triangles, which are base structures of many geometric shapes.

Coordinate System Angles

Angles in a coordinate system are positioned in a specific way for clarity.
An angle in standard position:

• Starts with its initial side on the positive x-axis.
• Its vertex is at the origin of the coordinate system (0,0).

This positioning helps consistently understand the angle's rotation and final placement. It's a universal method to easily identify angles in trigonometry and geometry which makes calculations more systematic and predictable.

Positive Rotation Direction

Understanding rotation direction is crucial while dealing with angles in the coordinate system. For positive angles, rotation is counter-clockwise.
This is the default for measuring angles starting from the positive x-axis.
Here’s how it works:

• Moving counter-clockwise means moving upwards on the circle, toward the positive y-axis, for angles between 0 to 180 degrees (0 to π radians).
• For a positive angle such as π/2 radians, you rotate 90 degrees counter-clockwise from the x-axis to the y-axis.

This direction is important because it maintains consistency across various calculations and diagrams.