Recreate the Unit Circle
Question: Where do the values of the trigonometric functions for the angles on the unit circle come from? Answer: They come from two “special” right triangles: the isosceles right triangle and the 30‐60‐90 right triangle. 2 1 30o 2 2 3 1 3 60o 45o 1 11 1
The isosceles right triangle gives The 30‐60‐90 right triangle gives the values for the values for 45 degrees. 30 degrees and for 60 degrees. Take a square of side 1 and draw a Take an equilateral triangle of side 2 and draw a diagonal. perpendicular bisector. The hypotenuse has length 2. So The bisector has length 3. So 2 2 1 3 3 sin 45° cos 45° sin 30° cos 30° tan 30° 2 2 2 2 3 tan 45° 1 and 3 1 sin 60° cos 60° tan 60° 3 2 2 Now build a table:
1. Write the numbers 0 –4 in a row. 01234 o o o o o 2. Write the angles 0, 30, 45, 60, and 90 degrees 0 30 45 60 90 beneath them. 3. For the values of sine, take the square roots 0 1 2 3 4 of the numbers in the top row and 4. Divide by 2: 1 2 3 sin θ 0 1 5. For the values of cosine, start with the 2 2 2 last number for sine and write the values 3 2 1 backwards cos θ 1 2 2 2 0 tan θ i sin θ cos θ 6. s just / 3 tan θ 0 3 1 3 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” For #2, use reference angles: ALL STUDENTS Every angle in quadrants 2, 3, and 4 have Quadrant 2 Quadrant 1 a corresponding reference angle in S stands for sine ALL are positive quadrant 1 whose trigonometric values Only sine is positive are the same, except for the sign. sin θ 0 sin θ 0 cos θ 0 cos θ 0 tan θ 0 tan θ 0 sin θ 0 sin θ 0 cos θ 0 cos θ 0 o o tan θ 0 tan θ 0 30 30 T stands for tangent C stands for cosine o o 30 30 Only tangent is positive Only cosine is positive Quadrant 3 Quadrant 4 TAKE CALCULUS
Now we can make the tables for the The figure shows that 30o is the reference other quadrants: angle for 150o, 210o, and 330o. Quadrant 2: Similarly, 60o is the reference angle for o o o o o o o Ө 120 135 150 180 120 , 240 , and 300 . o o o o Ref Ө 60 45 30 0 45o is the reference angle for 135o, 225o, 1 and 315o. sin θ 3 2 0 2 2 2 1 2 3 cos θ ‐1 Quadrant 4: 2 2 2 3 Ө 300 o 315o 330 o 360 o tan θ 3 1 0 3 o o o o Ref Ө 60 45 30 0 Quadrant 3: 3 2 1 sin θ 0 o o o o 2 2 2 Ө 210 225 240 270 o o o o 1 2 3 Ref Ө 3045 60 90 cos θ 1 2 2 2 1 sin θ 2 3 ‐1 3 2 tan θ 3 1 0 2 2 3 1 2 3 0 cos θ 2 2 2 tan θ 3 1 3 undef 3