Name: _________________________Date: _______________ Pd._____
Getting Started with Trigonometry and the Unit Circle Learning Task:
Part I: Angles as a Part of the Unit Circle
1. The circle below is called the unit circle. Why do you think it is call this? (Hint: look at the
radius of the circle)
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2. If angle bisectors are drawn from the right angles formed by the x and y axis (see dotted lines),
what angle measures result from these lines? ______ , ______ , ______ , ______
(Use a colored pencil to mark these lines all the same color)
3. Which of the angles have reference angles of 30o? ______ , ______, ______ , ______
(Use a colored pencil to mark these lines all the same color)
4. Which of the angles have reference angles of 60o?
______ , ______ , ______ , ______
(Use a colored pencil to mark these lines all the same color)
5. What is the angle measure when the terminal side of the angle lies on the negative side of the
x-axis? __________________________
6. What is the angle measure when the terminal side of the angle lies on the negative side of the
y-axis? __________________________
7. What is the angle measure when the terminal side of the angle lies on the positive side of the
y-axis? _________________________
8. For what angle measure(s) can the initial side and terminal sides overlap?
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9. Are these the only correct answers? Explain.
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Part II: Right Triangles and Coordinates on the Unit Circle
Recall rules on special right triangles:
45° - 45° - 90°
30° - 60° - 90°
45
30
𝑥√2
𝑥
2𝑥
𝑥√3
45
60
𝑥
𝑥
1. A 30o angle is marked on the circle. Label the point where the terminal side intersects the
circle as “A”. APPROXIMATE the coordinates of point A using the grid.
30o
Use the unit circle on the previous page.
2. Now, drop a perpendicular segment from the point you just put on the circle to the x-axis.
You should notice that you have formed a right triangle.
a) How long is the hypotenuse of your triangle? ____________
b) Using the rules of special right triangle on the previous page, find the lengths of the other
2 sides of your triangle in radical form and decimal form. Label these on your picture.
You now have the ACTUAL coordinates of point A.
Recall the Trigonometric Ratios:
𝑠𝑖𝑛𝑒 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑜𝑠𝑖𝑛𝑒 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
c) Using the sides of the triangle you created, find the sine and cosine of 30o in decimal
form.
sine 30o =
cosine 30o =
d) How are values of sine and cosine related to the coordinates of point A?
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3. Now reflect this triangle across the y-axis. Label the resulting image point as point B.
a) What are the actual coordinates of point B? _______________
b) How do these coordinates relate to the coordinates of point A?
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c) What obtuse angle was formed with the positive x-axis (the initial side) as a result of this
reflection? _________________
d) What is the reference angle for this angle? _____________________
4. Reflect the triangle into quadrant III so that there is a 30o reference angle.
a) What is the measure of the angle formed by the initial side?_______________________
b) Mark the lengths of each side. Label the point on the circle as C.
What are the coordinates point C? ___________________________
5. Reflect the triangle in the first quadrant over the x-axis.
a) What is the measure of angle made by initial side? ___________________
b) Mark the lengths of each side. Label the point on the circle as D.
What are the coordinates point D? ___________________________
6. Complete the table to summarize the results. Use special right triangle rules for exact
coordinates.

Quadrant
Exact
coordinates
(radical)
Decimal
coordinates
Notice that all of your angles so far have a reference angle of 30o.
7. What do you notice about the signs in relationship to the quadrants?
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