Unit Circle Exploration
Name: ________________________________
I) Special Right Triangles. Find the exact values of the other sides. Afraid.
1
Angle:
1
30°
45°
60°
sin()
cos()
Show all work to find:
A) tan(30°)
II) Unit Circle
B) tan(45°)
C) tan(60°)
Unit Circle Exploration
Name: ________________________________
 Write “1” as the hypotenuse length of each special right triangle cutout.
 Fill in the exact lengths of the other sides.
 Line up the cutouts on the unit circle to find specific coordinates along
the circumference of the circle. Label as many coordinates as possible.
DEFINITION:
Questions:
1. Find the exact value:
a) sin(150°) =
b) cos(225°) =
c) sin(300°) =
2. For which angles is sin() positive? Negative? Explain.
3. For which angles is cos() positive? Negative? Explain.
4. Tape your 30-60-90 triangle so that it points to 120°. Considering that a
side may be “positive” or “negative” (depending on direction),
a) What is the opposite side length?
b) What is the adjacent side length?
c) What is tan(120°)?
5. Tape your 45-45-90 triangle so that it points to 225°. Considering the signs
of the sides,
Unit Circle Exploration
Name: ________________________________
a) What is the opposite side length?
b) What is the adjacent side length?
c) What is tan(225°)?
6. For which quadrants is tan() positive? Negative? Explain.
7. Find the exact value:
a) sin(90°) =
b) cos(270°) =
c) sin(180°) =
8. Considering that a side length may be zero (in theory), in a rotation of 180°:
a) What is the length of the opposite (how tall)?
b) What is the length of the adjacent (how wide)?
c) What is tan(180°)?
9. Using similar logic, what is tan(90°)? Explain thoroughly.
10. How do you find the slope of a line?
Unit Circle Exploration
Name: ________________________________
11. Slope questions:
a) What is the slope of a horizontal line?
b) Which angle rotations result in a horizontal hypotenuse?
c) What is the slope of a perfectly diagonal line?
d) Which angle rotations result in a perfectly diagonal hypotenuse?
e) What is the slope of a vertical line?
f) Which angle rotations result in a vertical hypotenuse?
12. Is slope most closely related to sine, cosine or tangent? Explain.
13. Summarize. On the unit circle:
a) How is sine represented?
b) How is cosine represented?
c) How is tangent represented?