University:
Montgomery College
Course:
MATH 165: Precalculus
Academic year:

2020
Views:

132

Pages:

2
Author:

maselarmwpa

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 1 30o 2 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 2 1 cos 45° 2 2 sin 30° 1 2 cos 30° 3 tan 30° 2 3 3 1 tan 60° 2 3 and sin 60° 3 2 cos 60° 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 θ is 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 1 3 3 0 o 135 o 45 o 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