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Lesson Title: Extending Trig Functions to Larger Angles
Date: _____________ Teacher(s): ____________________
Course: Algebra II, Unit 7
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
F.TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all
real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
MP1:
MP2:
MP3:
MP7:
MP8:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Algebra II, Unit 7
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Have students, in pairs, pick any angle between 0
degrees and 90 degrees. Have pairs take their the angle
(call it  (theta)) and perform the following operations
on it:
180 – 
180 + 
360 – 
Using a calculator, have them find the sine, cosine and
tangent of their original angle and all three answers
they have computed.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
If the cos(5π/12) =0.2588, find three other angles that give
an answer of  0.2588 and indicate which are positive and
which are negative.
In what quadrant do you think the angle - π/6 would fall?
What do you think the answers to the sine and cosine
would be for this angle. Be prepared to justify your
answer tomorrow. (Look for evidence of MP1 and MP8.)
Now have them convert the angle measure for  into
radians and perform the following operations on it
π –
π +
2π–
Using a calculator, have them find the sine, cosine and
tangent of their original angle, given in radians, and all
three answers they have computed.
Ask them what is the same and different about their
answers. Point out that the angles they computed are
not angles that could be part of a right triangle. Ask
pairs to explain why. Allow for critique and
amendment. (Look for evidence of MP 2 and MP3.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Distribute the Activity 1 graph paper for drawing a unit circle, one to each pair of students. Have students, using
a compass, draw a circle with center at the origin and of radius one unit on the grid. Define this circle as the unit
circle. Ask pairs to construct an angle using the positive x-axis as one side and a ray from the origin intersecting
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Extending Trig Functions to Larger Angles
Course: Algebra II, Unit 7
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
with the circle in the first quadrant. The angles may be of different sizes. Ask pairs to estimate the measure of
their angle and the arc on the circle, both in degrees and radians. Also ask them to estimate the coordinates of the
point of intersection with the circle and label the point. (Look for evidence of MP 2 and MP3.)
2. Have pairs construct a triangle by drawing a vertical line from the point of intersection with the circle to the xaxis. Ask students to find the sine and cosine of the angle at the origin in two ways, using both their estimated
angle and a calculator and the ratios of the measures of the sides. (Make note that since the hypotenuse of the
right triangle is the radius of the circle, it is always equal to one and, therefore, the cosine and sine are merely the
coordinates of the point of intersection.) Ask them if their answers are the same or different. Ask them why they
should be the same or different. Allow for discourse among groups on this question. (They should be guided to
the realization that their answers should theoretically be the same, but, in reality, estimation error should make
them “slightly” different.) (Look for evidence of MP 2 and MP7.)
3. Have pairs draw another right triangle in the second quadrant, again with a vertex at the origin, on the unit circle
and on the x-axis. This triangle should have the exact same answers for the sine and cosine of the angle at the
origin. Have randomly selected pairs display their graphs and explain what rationale they used to create them.
Allow for critique and amendment. (Answers may involve reflection over the y-axis or simple changing of sign
of the x-coordinate.) Have pairs, similarly, construct triangles in the third and fourth quadrants. (Look for
evidence of MP1 and MP7.)
4. Explain that the angles for the triangles they are finding are used to find the trigonometric values of angles that
exceed π/2 radians we did in the warm-up. (The angles are measured starting at the positive x-axis moving
counterclockwise. The angle is typically measured in radians and then expands the possible angles that are in
the domain of sine and cosine to all real positive numbers. The answers become the actual coordinate values
of the point of intersection, positive or negative, depending on the quadrant. The answers repeat themselves as
they did for the angles on our unit circle and in the warm-up. If the trig ratios are written as a function, the values
in the domain that give similar answers can be identified. If one answer is known, other values can be found.
These values are positive and negative values of the same quantity, depending on the sign of the coordinate
values, as in the warm-up.)
5. Project the unit circle from the Activity 2 resource sheet on the board (or create a unit circle on the board or large
post-it paper). Assign each student with one trig function value and distribute a sticker. Have students indicate
whether their assigned trig angle value is positive or negative on the sticker with “+”/ “-”. Then invite the
students to estimate the location of their angle by placing the sticker on the unit circle that is on the board. Once
everyone has made their placement, as a class, discuss if any of stickers need to be adjusted. (Look for evidence
of MP1.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students will be able to convert between angle measures in degrees and radians.
Students will be able to find the sine and cosine of angles beyond π/2 using the concept of the unit circle.
They will complete a unit circle of angles and sine and cosine values successfully.
Evidence will be present during pairs and group work, pairs and group reporting, discussion of pairs and group
reporting, during the lesson closure activity, and on homework.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Vocabulary: unit circle, radian
Students may have trouble with the idea of an angle being greater than π or 1800. Stress on the unit circle as the
definition of the angle as opposed to only a triangle will help ameliorate this problem.
Students may also ask about the tangent of these angles. This can be addressed as the tangent being sine/cosine since
they are literally the opposite and adjacent side of the triangles created on a unit circle.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Extending Trig Functions to Larger Angles
Date: _____________ Teacher(s): ____________________
Course: Algebra II, Unit 7
Start/end times: _________________________
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Compass
Activity Sheet 1 and Activity Sheet 2
Blank stickers
There are many problem sets involving sine and cosine of
radian angles in textbooks.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Were groups able to successfully complete the unit circle?
Was there fluency on the completion of finding sine and cosine of angles related to reference (first quadrant) angles?
Were students able to correctly locate the angles on the unit circle?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.