Alg2 Lesson 4T.5: The Coordinates of Points on a Circle

Lesson 4T.5: The Coordinates

of Points on a Circle

Obj: I can identify the coordinates on the
unit circle.

EQ: How can I use the unit circle to identify
points and angles?

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Roles:
Facilitator
Scribe
Resourcer
Includer

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Trigonometry

Trigonometry is the branch of
mathematics that links two
fundamental geometric objects:
circles and triangles.

Part 1: Determining the

Coordinates of Points on the

Unit Circle

Determining a Point on a Circle

Is the angle θ closer to 1, 2, or 3 radians?

What are the coordinates of a circle in
terms of trigonometric expressions?

Estimate the value of sin(θ).

Estimate the value of cos(θ).

Handout 4T.5.A: Determining the Coordinates of
Points on the Unit Circle

Fun-Time Amusement Park has a
ride called the Sky Swing that lifts
riders into the air and spins them
counterclockwise over a water
ride called the Lazy River. The
park engineer knows that after the
rider rotates 0.5 radians from the
starting position, the rider will be
over the Lazy River.

Drag and Drop

Suppose the ride stops turning after it rotates radians. What are the coordinates of the point at which the rider is located when the ride stops turning? The coordinates of the location of the rider are (​

Placeholder

, ​

Placeholder

). In terms of the radius length, the horizontal distance of the rider from the vertical diameter is ​

Placeholder

and the vertical distance of the rider from the horizontal diameter is ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

cos(θ)

sin(θ)

Multiple Choice

Another amusement park has a swing ride with a radius of 45 feet. The ride stops after the rider
has rotated 2.5 radians from the starting position. What are the coordinates of the location of the rider in feet?

• (45cos(2.5),
  45sin(2.5))

• (45sin(2.5),
  45cos(2.5))

• (2.5cos(45),
  2.5sin(45))

• (2.5sin(45),
  2.5cos(45))

Part 2: Constructing Congruent

Reference Triangles

​

Insights from the diagrams

The values of sin(α) and sin(0.5) are equivalent. The first quadrant right triangle was formed using an angle of 0.5 radians with the positive x-axis and the second quadrant right triangle was formed using an angle of 0.5 radians with the negative x-axis. Furthermore, the radii (the hypotenuses of the right triangles) are congruent, which means the two right triangles are congruent and the vertical distance from the x-axis must be equal.

The values of cos(α) and cos(0.5) have opposite signs. The first quadrant right triangle was formed using an angle of 0.5 radians with the positive x-axis and the second quadrant right triangle was formed using an angle of 0.5 radians with the negative x-axis. Furthermore, the radii are congruent, which means the two right triangles are congruent. This means that the horizontal legs have the same length, but the second quadrant right triangle extends to the left. We indicate this by making the value of the cos(α) negative.

​

Angle Relationships

​

More Angle Relationships

Handout 4T.5.B: Exploring Reference Triangles

For Practice: https://tinyurl.com/mwxpvce4

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin
the wheel as a class and spend a minute or so
discussing our answers.