Download Recreate the Unit Circle: 0 1 2 3 4 60 30 0 90 45 sinθ 0 1 cosθ 1 0 0

Recreate the Unit Circle:
Question: Where do the values of the trigonometric functions for the angles on the unit circle come from?
They come from two “special” right triangles: the isosceles right triangle and the 30‐60‐90 right triangle. Answer:
2
2
1
30o
1
2
3
3
60o
1
1
45o
1
1
The isosceles right triangle gives the values for 45 degrees.
The 30‐60‐90 right triangle gives the values for 30 degrees and for 60 degrees.
Take a square of side 1 and draw a diagonal.
Take an equilateral triangle of side 2 and draw a perpendicular bisector.
The hypotenuse has length 2. So
The bisector has length 3. So
sin 45°
tan 45°
2
cos 45°
2
1
2
2
sin 30°
sin 60°
1
cos 30°
2
and
3
tan 30°
2
3
3
1
tan 60°
2
3
3
cos 60°
2
Now build a table:
0
1. Write the numbers 0 – 4 in a row.
o
2. Write the angles 0, 30, 45, 60, and 90 degrees 0
beneath them.
3. For the values of sine, take the square roots 0
of the numbers in the top row and
4. Divide by 2:
sin θ 0
5. For the values of cosine, start with the last number for sine and write the values backwards
cos θ 1
6. tan θis just sin θ / cos θ
tan θ
0
1
30
2
o
45
3
o
60
4
o
1
2
3
1
2
2
2
3
2
3
2
2
2
1
2
3
3
1
3
90
o
4
1
0
undef
To complete the values for the unit circle, we need two more things:
1. The signs of the trigonometric functions in different quadrants
2. The angles in different quadrants that have 30, 45, and 60 degrees as reference angles
For #1, remember “All Students Take Calculus”
ALL
STUDENTS
For #2, use reference angles:
Every angle in quadrants 2, 3, and 4 have a corresponding reference angle in quadrant 1 whose trigonometric values are the same, except for the sign.
Quadrant 1
Quadrant 2
ALL are positive
S stands for sine
Only sine is positive
sin θ 0
cos θ 0
tan θ 0
sin θ
cos θ
tan θ
0
0
0
sin θ
cos θ
tan θ
0
0
0
sin θ
cos θ
tan θ
0
0
0
30
C stands for cosine
Only cosine is positive
T stands for tangent
Only tangent is positive
Quadrant 3
30
o
30
o
30
o
o
Quadrant 4
CALCULUS
TAKE
Now we can make the tables for the other quadrants:
The figure shows that 30o is the reference angle for 150o, 210o, and 330o.
Quadrant 2:
Similarly, 60o is the reference angle for 120o, 240o, and 300o.
o
o
o
Ө
120
Ref Ө
60
sin θ
3
2
2
2
1
2
cos θ
1
2
2
2
3
2
‐1
tan θ
3
3
3
0
o
135
o
45
o
1 150
30
o
180
0
o
0
Quadrant 4:
Quadrant 3:
Ө
Ref Ө
210
30
o
o
o
225
45
o
240
60
o
o
270
90
45o is the reference angle for 135o, 225o, and 315o.
o
o
sin θ
1
2
2
2
3
2
‐1
cos θ
1
2
2
2
3
2
0
tan θ
3
3
1 3
undef
Ө
300 o
o
315o
o
330 o
o
360 o
Ref Ө
60
sin θ
3
2
2
2
1
2
0
cos θ
1
2
2
2
3
2
1
tan θ
3
1 3
3
0
45
30
0
o