Alg2 Lesson 4T.6: Common Right Triangles in a Unit Circle

Lesson 4T.6: Common Right

Triangles in a Unit Circle

Obj: I can simplify expressions using
trigonometric identities.

EQ: What is the relationship between
angles and points on the unit circle?

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…

Part 1: Exploring the
Pythagorean Theorem

Multiple Choice

Which of the following is equivalent to a? (let c = 1)

• a = tan(55°)

• a = sin(55°)

• a = cos(55°)

• a = sec(55°)

• a = csc(55°)

Multiple Choice

Which of the following is equivalent to b? (let c = 1)

• b = tan(55°)

• b = sin(55°)

• b = cos(55°)

• b = sec(55°)

• b = csc(55°)

Observations

It appears that the sum of the squares of sine and cosine of an angle is the same regardless of the angle measure.

The sum of the squares of the sine and cosine of any angle is equal to 1.

To prove this, we could write the equation using a general angle, like θ, and determine if the Pythagorean relationship is true.

Record and Discuss what you notice in this image

Fill in the Blank

What is the length of the hypotenuse of the reference triangle of the unit circle?

Drag and Drop

The signed vertical distance of a point on a unit circle from the horizontal
axis is given by the ​

Placeholder

of the associated angle. The signed horizontal distance of a point on a unit circle from the
vertical axis is given by the ​

Placeholder

of the associated angle. This can be represented by the ordered pair (​

Placeholder

, ​

Placeholder

)

Drag these tiles and drop them in the correct blank above

sine

cosine

cos(θ)

sin(θ)

Multiple Choice

How can we use the Pythagorean theorem to express the relationship between
the legs of the reference triangle of the unit circle, in terms of the sine and the cosine of the
associated angle?

• sin²(θ) + cos²(θ) = 1
• sin²(θ) * cos²(θ) = 1
• sin²(θ) + cos²(θ) = 2
• sin²(θ) - cos²(θ) = 1

Part 2: Investigating Right
Triangles in a Unit Circle

Reference Angles & Triangles

A reference triangle is drawn so that the horizontal leg lies on the positive or negative x-axis and the vertical leg is perpendicular to the x-axis.

A reference triangle in quadrant II, III, or IV is congruent to a reference triangle in quadrant I. They have the same reference angle. The triangle in
quadrant II, III, or IV is a reflection of the quadrant I triangle over one or both axes.

A reference angle is the acute angle that the terminal ray of the angle in standard position makes with the negative x-axis in quadrants II and III and with the positive x-axis in quadrants I and IV.

Jigsaw Activity

There will be 3 groups. After completing your
group’s handout, another group will be
formed containing at least 1 member from
each of the initial 3 groups. Be prepared to
explain your handout to your new group.

Handout 4T.6.D: Common Angles in a Unit Circle

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.