Questions: Introducing the Unit Circle Example #2: 45 Degree Reference Angle Using Triangles in the Unit Circle: A unit circle is graphed on the coordinate plane. - Notice that a line segment can be drawn from any point on the unit circle with coordinates (x, y) perpendicular to the x-axis. - This line segment will form a right triangle with the radius as the hypotenuse. In this section, you will use special right triangles to show that any point on the unit circle satisfies the equation of the unit circle. Enter the coordinates (x, y) of the point shown on the coordinate plane, using the exact values of x and y. If your answer is correct, you will see "(0" next to your answer. Submit

Introducing the Unit Circle

Example #2: 45 Degree Reference Angle

Using Triangles in the Unit Circle:
A unit circle is graphed on the coordinate plane.
- Notice that a line segment can be drawn from any point on the unit circle with coordinates (x, y) perpendicular to the x-axis.
- This line segment will form a right triangle with the radius as the hypotenuse.

In this section, you will use special right triangles to show that any point on the unit circle satisfies the equation of the unit circle.

Enter the coordinates (x, y) of the point shown on the coordinate plane, using the exact values of x and y.

If your answer is correct, you will see "(0" next to your answer.

Submit

Transcript text: Introducing the Unit Circle Example #2: 45 Degree Reference Angle Using Triangles in the Unit Circle: A unit circle is graphed on the coordinate plane. - Notice that a line segment can be drawn from any point on the unit circle with coordinates $(x, y)$ perpendicular to the x-axis. - This line segment will form a right triangle with the radius as the hypotenuse. In this section, you will use special right triangles to show that any point on the unit circle satisfies the equation of the unit circle. Enter the coordinates $(x, y)$ of the point shown on the coordinate plane, using the exact values of $x$ and $y$. If your answer is correct, you will see "(0" next to your answer. $\square$ Submit

Solution

Solution Steps

Step 1: Identifying the given information

We are given a unit circle (radius 1) and a 45-degree angle. We need to find the coordinates (x,y) of the point where the angle intersects the circle.

Step 2: Using the properties of a 45-45-90 triangle

A 45-degree angle in a unit circle creates a 45-45-90 triangle. In a 45-45-90 triangle, the two legs are equal in length, and the hypotenuse is √2 times the length of each leg. Since the hypotenuse (radius) is 1, the legs (x and y coordinates) are each 1/√2.

Step 3: Rationalizing the denominator

We simplify 1/√2 by multiplying the numerator and denominator by √2, resulting in √2/2.