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
3
3
1
45o
2
60o
1
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
1
cos 30°
2
and
sin 30°
3
cos 60°
2
sin 60°
1
2,
3
30o
1
tan 60°
2
3
Now we can label the points (cos , sin for the first quadrant
2
2,
2
3
45o
3
3
2
2
60o
3
tan 30°
2
2,
1
2
1
2
2
2,
3
2,
2
2
120o
135o
3
To get the values for the 2nd
quadrant, imagine flipping the triangles about the Y‐
axis
2,
1
150o
2
We should get almost the same values as before, but the x values should be negative
210o
3
2,
2
1
225o
2
240o
2
2,
2
1
3
2,
3
330o
2
2,
1
2
315o
300o
2
For the third quadrant, flip about the Y‐axis 1
Now both x and y values will be negative
2,
3
2,
2
2
2
Finally, for the fourth quadrant, flip again about the Y‐axis Now only the y values will be negative
1
2
3
2,
2
2,
1
3
2,
2
2
1
3
2
2
120o
135o
60o
150o
2
2,
2
2,
3
45o
2
2,
1
Putting it all together gives us
2
30o
The Unit Circle
210o
3
2,
2
1
2,
2
315o
240o
2
2
300o
2
1
2,
3
330o
225o
3
2
1
2,
2,
3
2
2,
2
1
2
2