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Algebra 2 Unit 2 Trigonometry Performance Task 3 (Alternate)
Open the TI-Nspire document The_Unit_Circle.tns.
In this activity, you will click on a slider to change the measure of the central angle
and create a series of line segments that determine the graph of a sinusoidal function.
Move to page 1.2.
1. The circle pictured is called a unit circle. Why is that term used?
2. Use the slider to make three segments appear. What is the relationship between the right triangle in the unit circle
and the vertical line segments?
3. Will the lengths of the line segments continue to increase? Why or why not?
4. Use right triangle trigonometry to explain the relationship between the angle  and the highlighted leg of the right
triangle in the unit circle. What trigonometric function can be represented by the length of the leg of the right
triangle?
5. Use the slider until you obtain values of  such that  <  <
3
. Explain the placement of the line segments.
2
6. Continue to use the slider until  = 2 to graph a continuous function. What do the coordinates of the points on the
continuous function represent?
7. If we continued to graph the function for values of  such that 2 <  < 4, describe what you would expect to see.
Explain your reasoning.
Move to page 2.2.
8. Use the slider until three segments appear. What is the relationship between the right triangle in the unit circle and
the vertical line segments?
9. Use right triangle trigonometry to explain the relationship between the angle  and the highlighted leg of the right
triangle in the unit circle. What trigonometric function can be represented by the length of the leg of the right triangle?
10. Use the slider until you obtain values of  such that

<  < . Explain the placement of the line segments.
2
11. Continue to use the slider until  = 2 to graph a continuous function. What do the coordinates of the points on the
continuous function represent? What does the x-coordinate represent? What does the y-coordinate represent?
Answer the following questions based on the explorations you have done with the calculator. You may refer back to
the calculator as needed.
Use your sine curve on page 1.2.
12. a) What is the period of the sine curve? That is, what is the wavelength? ___________________________
b) After how many radians does the graph start to repeat?______________________________________
c) How do you know it repeats after this point?
13. What are the zeroes of this function? (Remember: The x-values are measuring angles and zeros are the x-intercepts.)
14. What are the x-values at the maxima and minima of this function?
15. What are the y-values at the maxima and minima?
16. Imagine this function as it continues in both directions. Explain how you can predict the value of the sine of 390°.
17. Explain why sin 30° = sin 150°. Refer to both the unit circle and the graph of the sine curve.
ADVANCED
Write a paragraph comparing and contrasting the sine and cosine curves. You may use the questions below
to help guide your paragraph.




In what ways are the sine and cosine graphs similar? Be sure to include a discussion of intercepts, maxima,
minima, and period.
In what ways are the sine and cosine graphs different? Again, be sure to include a discussion of intercepts,
maxima, minima, and period.
Will sine graphs continue infinitely in either direction? How do you know? Identify the domain and range of
y = sin x .
Will cosine graphs continue infinitely in either direction? How do you know? Identify the domain and range of
y = cos x .
Scoring Guide Task 3:
Advanced
 Meets all of the
“Proficient” criteria
plus:
 Completes advanced
paragraph comparing
and contrasting sine
and cosine.
Proficient
 Accurately completes questions
related to sine curve (1-7).
 Accurately completes questions
related to cosine curve (8-11).
 Accurately completes follow-up
questions (12-17)
Progressing
 Meets 1 - 2 of the
Beginning
 Meets fewer than 1
“Proficient” criteria
of the “Proficient”
(75% accuracy on
criteria
(60% accuracy or less
on questions)
questions)