Defining the Unit Circle: Videos & Practice Problems

The unit circle is a circle with a radius of 1, centered at the origin, crucial for understanding trigonometric functions. Angles are measured in degrees and radians, with key points at (1,0), (0,1), (-1,0), and (0,-1). The equation of the unit circle is $x^2+y^2=1$. Points on the circle correspond to specific angles, facilitating the verification of coordinates using this equation.

Introduction to the Unit Circle

Understanding the unit circle is crucial for solving various trigonometric problems. The unit circle is defined as a circle with a radius of 1, centered at the origin (0, 0) of a coordinate system. It encompasses all angles from 0 to 360 degrees, which can also be expressed in radians, where a full rotation of 360 degrees corresponds to \(2\pi\) radians.

Key angle measures on the unit circle include:

• 0 degrees = 0 radians
• 90 degrees = \(\frac{\pi}{2}\) radians
• 180 degrees = \(\pi\) radians
• 270 degrees = \(\frac{3\pi}{2}\) radians
• 360 degrees = \(2\pi\) radians

Each angle on the unit circle corresponds to specific coordinates (x, y). For instance:

• At 0 degrees (0 radians), the coordinates are (1, 0).
• At 90 degrees (\(\frac{\pi}{2}\) radians), the coordinates are (0, 1).
• At 180 degrees (\(\pi\) radians), the coordinates are (-1, 0).
• At 270 degrees (\(\frac{3\pi}{2}\) radians), the coordinates are (0, -1).

The equation of the unit circle can be derived from its definition. Since the center is at the origin and the radius is 1, the equation is given by:

\[x^2 + y^2 = 1\]

This equation indicates that any point (x, y) on the unit circle will satisfy this relationship. To determine if a point lies on the unit circle, one can substitute the coordinates into this equation. If the equation holds true, the point is on the unit circle.

For example, to check if the point (1, 1) is on the unit circle, we substitute:

\[1^2 + 1^2 = 1 + 1 = 2 \quad (\text{not on the unit circle})\]

Conversely, for the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), we check:

\[\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1 \quad (\text{on the unit circle})\]

This point corresponds to an angle of 60 degrees or \(\frac{\pi}{3}\) radians. As you continue to explore the unit circle, you'll find that each point corresponds to a specific angle, reinforcing the connection between trigonometric functions and their geometric representations.

As you practice, remember that mastering the unit circle will greatly enhance your understanding of trigonometry and its applications.

Identify the quadrant that the given angle is located in.
$\frac{7\pi}{4}$ radians

• A. Quadrant I
• B. Quadrant II
• C. Quadrant III
• D. Quadrant IV

Answer: (D)

Identify the quadrant that the given angle is located in.
$\frac{\pi}{7}$ radians

• A. Quadrant I
• B. Quadrant II
• C. Quadrant III
• D. Quadrant IV

Answer: (A)

Identify the quadrant that the given angle is located in.
$\frac{2\pi}{3}$ radians

• A. Quadrant I
• B. Quadrant II
• C. Quadrant III
• D. Quadrant IV

Answer: (B)

Identify the quadrant that the given angle is located in.
$\frac{6\pi}{5}$ radians

• A. Quadrant I
• B. Quadrant II
• C. Quadrant III
• D. Quadrant IV

Answer: (C)

Test whether the point is on the unit circle by plugging it into the equation, $x^2+y^2=1$.
$\left(\frac{-\sqrt2}{2},\frac{-\sqrt2}{2}\right)$

• A. On Unit Circle
• B. NOT on Unit Circle
• C. More information is required

Answer: (A)

Here’s what students ask on this topic:

What is the unit circle and why is it important in trigonometry?

The unit circle is a circle with a radius of 1, centered at the origin (0,0) on the coordinate plane. It is crucial in trigonometry because it provides a geometric representation of the trigonometric functions sine, cosine, and tangent. Each angle on the unit circle corresponds to a specific point (x, y), where x is the cosine of the angle and y is the sine of the angle. This helps in understanding and solving trigonometric problems, as well as converting between degrees and radians.

How do you convert degrees to radians on the unit circle?

To convert degrees to radians on the unit circle, use the conversion factor π radians = 180 degrees. The formula is:

$\frac{\theta \pi }{180}$

For example, to convert 90 degrees to radians:

$\frac{90\pi }{180}=\frac{\mathrm{\pi }}{2}$

What is the equation of the unit circle?

The equation of the unit circle is derived from the general equation of a circle, $x^2+y^2=r^2$, where r is the radius. For the unit circle, the radius r is 1, so the equation simplifies to:

$x^2+y^2=\; 1$

This equation means that any point (x, y) on the unit circle will satisfy this relationship.

How do you verify if a point is on the unit circle?

To verify if a point (x, y) is on the unit circle, plug the coordinates into the unit circle equation $x^2+y^2=\; 1$. If the equation holds true, the point is on the unit circle. For example, for the point (1/2, √3/2):

$\frac{1^2}{4}+\frac{{\sqrt{3}}^{}}{2}=\; 1$

Calculating this gives:

$\frac{1}{4}+\frac{3}{4}=\; 1$

Since 1 = 1, the point is on the unit circle.

What are the key points on the unit circle and their corresponding angles?

The key points on the unit circle and their corresponding angles are:

• (1, 0) at 0 degrees or 0 radians
• (0, 1) at 90 degrees or π/2 radians
• (-1, 0) at 180 degrees or π radians
• (0, -1) at 270 degrees or 3π/2 radians

These points help in understanding the trigonometric functions and their values at these specific angles.