Download 4.3 Unit circle notes Vocab you must know 1st: initial side, vertex

4.3 Unit circle notes
Vocab you must know 1st: initial side, vertex, terminal side, measure of an angle, positive angles, negative
angles, standard position, co-terminal angles, trig functions of any angle, reference triangle, quadrantal
angles, trig functions of real numbers, circular functions, periodic function
Terminal side, initial side, vertex:
Standard position – vertex at origin
and initial side is on the positive x-axis
positive & negative angles:
coterminal angles:
2 positive coterminal angles
positive & negative coterminal
Trig functions of any Angle: Let θ be any angle in standard position and let P(x, y) be any point on the
terminal side of the angle. Let r denote the distance from P(x, y) to the origin (let r =
). Then
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
reference triangle: (when either a 45-45-90 or 30-60-90 triangle is formed)
quandrantal angles: angles whose terminal side lies along the x-axis or y-axis
Unit circle: a circle with radius 1 w/ center (0, 0)
Trig functions of real numbers: Let b be any number, and let P(p, q) be the point corresponding to m
when the number line is a wrapped onto the unit circle as described below. Then
sin b = q
cos b = p
tan b =
csc b =
sec b =
cot b =
Periodic functions: A function y = f(t) is periodic if there is a positive number c such that f(t + c) = f(t)
for all values of t in the domain of f. The smallest such number c is called the period of the functions.
A) point P (a, b) is on the terminal side of angle θ. Evaluate the 6 trig functions for θ. If the function is
undefined, write “undefined”
Steps:
1. Draw a coordinate plane and plot the point
2. Draw a triangle and label your points a and b correctly
3. Use Pythagorean theorem to find r
4. Find all 6 trig ratios (if the triangle is in the 2nd, 3rd,
or 4th quadrant you can have a “-“ answer)
ex: 1. P (-4, -6)
2. P (-3, 0)
B) evaluate without a calculator by using ratios in a reference triangle
Steps:
1. Convert to degrees if necessary
2. Draw the angle in the coordinate plane
3. Determine if a 45-45-90 or 30-60-90 triangle is formed
4. Evaluate the ratio, remember the x-value is always 1 (use the
standard 45-45-90 and 30-60-90 given the other day) –
don’t forget “negative” when necessary
Ex: 3. tan 300˚
4.csc
5. cos
C) use quadrantal angles to find sin, cos, and tan. If the function is
undefined, write “undefined”
Steps:
1. Determine where the point is (it should end on one of the axis)
2. Determine the coordinates of P
3. Use “definition of trig functions in any angle” to find the ratio,
sin = y/r, cos = x/r, tan = y/x
Ex: 6. -270˚
7.
D) evaluate without a using a calculator
Steps:
1. Determine which quadrant(s) would satisfy the conditions given, then choose one of the quadrants
(it won’t matter which one you choose)
2. Draw and label your triangle in the chosen quadrant (the origin will always be θ)
3. Use Pythagorean theorem to find the 3rd side of the triangle
4. Use trig ratios to find the ones that are being asked for. Don’t forget “-“ if the triangle is in
quadrant 2 or 4
Ex: 8. Find cos θ and cot θ if sin θ = and tan θ < 0 (think where would y/x be negative)
9. find csc θ and cot θ if tan θ = -4/3 and sin θ > 0 (think where is y positive)
E) find the value of the unique real number θ between 0 and 2 that satisfies the 2 given conditions
Steps:
Need to use the Unit circle for these
1. Find where the 1st given trig ratio is (there will be 2 points)
2. Determine which point would satisfy the 2nd condition, the coordinating radians is your answer
Ex: 10. sin θ =
11. cos θ =
and tan θ > 0
and sin θ < 0
Homework:
p. 347-349 #7-11 odd, 25-47 odd, 67-69 odd