Problem 15 - Precalculus : Building Concepts and Connections

Question

Sketch the angles in standard position. $$\frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

The angle \(\frac{7 \pi}{4}\) radians in standard position will lie in the fourth quadrant of the unit circle, 45° clockwise from the positive x-axis.

Step 1: Understanding the Concept

The notation \(\frac{7 \pi}{4}\) indicates an angle measured in radians. Since \(2\pi\) radians amount to one full circle (360 degrees), \( \frac{\pi}{4}\) can be understood to be 45 degrees. Thus, \( \frac{7 \pi}{4}\) equals \(7 \times 45\) degrees or 315 degrees.

Step 2: Sketching the Unit Circle

Draw a circle with the origin '(0,0)' at its center. Set up the x and y axes, with positive directions to the right and upward and negative directions to the left and downward respectively.

Step 3: Marking the Angle

Starting from the positive x-axis (the initial side), mark an angle of 315 degrees or \(\frac{7 \pi}{4}\) radians in the counterclockwise direction. The terminal side of the angle will lie in the fourth quadrant of the unit circle.

Key Concepts

Radian Measure

Radians are a unit of angular measurement. Unlike degrees, radians provide a more natural way of expressing angles, particularly in mathematics. One full circle is equivalent to \(2\pi\) radians, just like it is equivalent to 360 degrees. This means that the relationship between degrees and radians is straightforward:

• 180 degrees = \(\pi\) radians
• 90 degrees = \(\frac{\pi}{2}\) radians
• A smaller segment, like \(\frac{\pi}{4}\), equals 45 degrees

To convert from degrees to radians, use the formula: \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). For example, \(\frac{7\pi}{4}\) radians corresponds to \(7 \times 45 = 315\) degrees.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with its center at the origin of a coordinate plane and a radius of 1 unit. The unit circle allows us to define trigonometric functions for all angles:

• 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 standardized circle helps in understanding how angles correspond to points on a plane.
As you measure angles around the circle from the positive x-axis, the movement helps visualize angles in radians and degrees.

Standard Position Angles

Angles in standard position have their vertex at the origin and their initial side on the positive x-axis. This positioning offers a reliable way to visualize and measure angles based on their direction and magnitude.
An angle can be measured:

• In a counterclockwise direction, which is positive.
• Clockwise direction gives a negative measurement.

For example, the angle \(\frac{7\pi}{4}\) radians is positioned starting from the positive x-axis and moving counterclockwise, landing in the fourth quadrant of the unit circle.

Quadrants of the Unit Circle

The unit circle is divided into four areas known as quadrants. Each quadrant helps specify the signs of sine and cosine values for angles.

• First Quadrant - angles from 0 to \(\frac{\pi}{2}\) radians, both sine and cosine are positive.
• Second Quadrant - angles from \(\frac{\pi}{2}\) to \(\pi\) radians, sine is positive, cosine is negative.
• Third Quadrant - angles from \(\pi\) to \(\frac{3\pi}{2}\) radians, both sine and cosine are negative.
• Fourth Quadrant - angles from \(\frac{3\pi}{2}\) to \(2\pi\) radians, sine is negative and cosine is positive.

Understanding these quadrants helps locate the terminal side of any angle, such as \(\frac{7\pi}{4}\) radians, which resides in the fourth quadrant.