Algebra II/Trigonometry: Angles and the Unit Circle
Quadrant 2 (counterclockwise) degrees
-90° to 180° -π/2 radians to π radians -but since something can around a circle many times it cold also be an angle like 510° which would still be in quadrant 1 because its terminal side would be in quadrant 2 -we determine what quadrant an angle is in by where the terminal side ends
a circle has 360 degrees or _________ radians
2π radians -we know this because circumference is one way around the circle and that is the same idea for degrees
how many intervals are there between 0 and the period on the x axis
4
the sine function and the unit circle
essentially, instead of just overlapping on itself the sine function takes the unit circle and extends its y coordinates
initial side
the positive x axis
Quadrant 3 (clockwise) degrees
- (-180°) to (-270°) - -π radians to -3π/2 radians (both are negative) -but since something can around a circle many times it cold also be an angle like 660° which would still be in quadrant 1 because its terminal side would be in quadrant 4 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 4 (clockwise) degrees
- (-270°) to (-360°) - -3π/2 radians to -2π radians (both are negative) -but since something can around a circle many times it cold also be an angle like 660° which would still be in quadrant 1 because its terminal side would be in quadrant 4 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 2 (clockwise) degrees
- (-90°) to (-180°) - -π/2 radians to -π radians (both are negative pi) -but since something can around a circle many times it cold also be an angle like 660° which would still be in quadrant 1 because its terminal side would be in quadrant 4 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 1 (clockwise) degrees
-0° to -90° -0 radians to -π/2 radians -but since something can around a circle many times it cold also be an angle like 660° which would still be in quadrant 1 because its terminal side would be in quadrant 4 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 1 (counterclockwise) degrees
-0° to 90° -0 radians to π/2 radians -but since something can around a circle many times it cold also be an angle like 405° which would still be in quadrant 1 because its terminal side would be in quadrant 1 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 3 (counterclockwise) degrees
-180° to 270° -π radians to 3π/2 radians -but since something can around a circle many times it cold also be an angle like 570° which would still be in quadrant 1 because its terminal side would be in quadrant 3 -we determine what quadrant an angle is in by where the terminal side ends
Quadrant 4 (counterclockwise) degrees
-270° to 360° -3π/2 radians to 2π radians -but since something can around a circle many times it cold also be an angle like 660° which would still be in quadrant 1 because its terminal side would be in quadrant 4 -we determine what quadrant an angle is in by where the terminal side ends
what is the circumference of a circle
Circumference = 2πR
what is the circumference of the unit circle
Circumference in general is 2πR but since the radius of a unit circle is 1 all we need is 2π
angle variables
Beta, alpha, theta, gamma
can the period be negative
NO, the period can never be negative because the period measures distance and we can never have negative distance
unit circle
a circle of radius 1 whose center is at the origin -the picture is what it looks like when you put the unit circle on the graph -when you make an angle on the unit circle you follow the same pattern as everything above: you start at standard position and then just move around the circle either clockwise or counterclockwise
what is a in y = a sin(b∅)
a is always positive and it is the amplitude so it tells us how high and low the sine function goes e.g. if |a| = 4 then the highest the sine function will go (its highest y value) will be 4 and the lowest it will go will be (-4)
what is a in y = a cos(bx)
a is the amplitude -if a is negative then the graph start below the x axis and if its positive it starts above -the amplitude is where the cosine graph will always begin
what is the pattern of a cosine function
amplitude, x intercept, amplitude, x intercept, amplitude -these amplitudes will be maxes or mins based on what the amplitude is (if it is positive or negative)
what is b in y = a cos(bx)
b counts the number of cycles of the cosine function between 0 and 2π
what does the period of a function tell you
calculated using 2π/b you can find the last x value in one complete cycle of a function which can then allow you to find the subintervals
clockwise rotation produces ____________ angles
clockwise rotation produces negative angles
what are the (x,y) coordinates of where the terminal side intersects with the unit circle
cosin of angle = x sin of angle = y (cosin, sin) = (x,y)
counter clockwise rotation produces ________________ angles
counter clockwise rotation produces positive angles
what is b in y = a sin(b∅)
counts how many complete cycles there are between 0 and 2π on the graph
Degree to radian conversion factor
degree * π/180 e.g. 45° * π/180 = π/4 radians
period can also tell you
how fast or slow a circle is using because we will move into circles of varying sizes but right now we are only using the unit circle
amplitude
how high or how low the function goes -minium/maximum height -is always positive
finding coterminal angles with radians
if you leave the radians in radians, all you have is: -to do to find a positive coterminal angle is add 2π to what you have and that answer will be the positive coternminal angle -to find a negative coterminal angle minus 2π from what you have and that answer will be the negative conterminal angle if want to convert it to degrees just use the conversion factor, multiply the radians by 180/π, then add 360 or minus 360 from that degree value to get a positive or negative coterminal angle
trigonometry
input rotation, output location
when the a value of y = a sin(b∅) is positive what does that mean
it means that you start you function above the x axis for both positive and negative cycles
when the a value of y = a sin(b∅) is negative what does that mean
it means that you start your function below the x axis for both positive and negative cycles
what does rotation mean
it means we start at standard position which creates the initial side and then we rotate either clockwise or counterclockwise around the x y plane into the 4 quadrants
where does the cosine function start
it starts at the amplitude -below or above the x axis -the coordinates are (0, amplitude) e.g. if the cosine function is y = 4cos(3x) -that means that the amplitude is 4 which is positive so the consine function will start at (0,4) and that also means that (#, -4) will be its minimum
what does a trig function do
it takes a rotation and it outputs a location on the unit circle
what does the sine function do
it unwraps the y coordinates of the unit circle onto the x,y coordinate plane which spreads them along the x axis -it matches the measure ∅ (like x) of an angle in standard position with the y coordinate of a point on the unit circle -it takes the top of the circle and the bottom of the circle and essentially spreads it out over the x axis
is there a limit to how much rotation there is
no it can just keep rotating a certain number of degrees
does the initial side change based off of whether it is rotating clockwise or counter clockwise
no the initial side will stay the same the only things that change are the degree numbers for the quadrants the direction of rotation -the initial side stays the same and it stays a 0° at the start
cycle
one complete pattern of the trig function
ray
one endpoint and extends without end in one direction only
circumference is a
perimeter measurement which means it's a 1st degree measurement
how to calculate the period
period uses b from y = a sin(b∅)so: period = 2π/b
vertex
point where two lines, line segments, or rays meet e.g. the origin is always the vertex when it is in standard position
Radian to degree conversion factor
radian * 180/π e.g. 7π * 180/π = 315°
what is range
set of y values -for sine and socine the y values oscillate between the amplitudes (which are y values)
where do we get 180/π from
since a circle is 360 and 2π radians, if we divide each by two we get half a circle is 180 degrees and π radians
how to calculate subintervals
subintervals uses the period so: sub = period * 1/4
what do we call: y coordinate of where the terminal side of an angle intersects with the unit circle / x coordinate of where the terminal side of an angle intersects with the unit circle
tan
tan (x) =
tan (x) = sin (x) / cos(x) akak -tan (x) = y / x
the cosine function unwraps the _____ coordinates of the unit circle
the cosine function unwraps the x coordinates of the unit circle
period
the horizontal distance of one cycle -is calculated using 2π/b
all angles will start at
the initial side which has a degree measure of 0° or 0 radians
where does the sine function start
the origin -(0,0)
the sine function unwraps the _____ coordinates of the unit circle
the sine function unwraps the y coordinates of the unit circle
what does the x axis become with the sine function
the x axis means time -not really but its helpful for visualization
cosin is
the x coordinate of the point where the terminal side of the angle interesects with the unit circle e.g. cosin(90) = 0 -this is because for the unit circle, a degree of 90 means the terminal side intersects with the unit circle at (0, 1) and since cosin is the x coordinate of that intersection our answer would be 0 because that is the x coordinate
sin is
the y coordinate of the point where the terminal side of the angle interesects with the unit circle e.g. sin(90) = 1 -this is because for the unit circle, a degree of 90 means the terminal side intersects with the unit circle at (0, 1) and since sin is the y coordinate of that intersection our answer would be 1 because that is the y coordinate
coterminal angles
they are angles, both of which start from standard position, that share a terminal side -there are positive coternminal angles (you add 360 from given angle) -there are negative coterminal angles (you substract 360 from given angle) -there are an infinite amount of coterminal angles for any given angle because you just add or subtract an infinite amount of circles (360°) to it -angles that share a terminal side, and differ by a multiple of 360° meaning you add or subtract 360 at least once from an angle e.g. 32° 32°+360° = 392° and 32° - 360° = -328° -for the angle 32°, a positive coterminal is 392° and a negative coterminal is -328°
what are subintervals
they are the intervals between the period of the function and 0 -you calculate them by doing: period * 1/4 -once you get this number you know it is the first interval between 0 and the first tick on the graph and then you just add that number onto itself until you get to the period
trigonmetric functions are called periodic functions because
they repeat
will this graph start below or above the x axis (it will always initially start at the origin) y = - sin (-x)
this graph will start above the x axis -even though we have a negative amplitude and that usually means that we start below the x axis, since the b value is negative, for a sine function which is an odd function that is like multiplying a negative by a negative which will give us a positive -that new positive value is why we start about the x axis
finding cosin with radians
this is the exact same process as finding it with degrees -all you have to do is right the x coordinate of where the terminal angle intersects with the unit circle e.g. cosin(3π/2) = 0 -this is because the coordinates where they intersect is (0,-1) and all you need to do is write the x coordinate which in this case is just 0
finding sin with radians
this is the exact same process as finding it with degrees -all you have to do is right the y coordinate of where the terminal angle intersects with the unit circle e.g. sin(3π/2) = =1 -this is because the coordinates where they intersect is (0,-1) and all you need to do is write the y coordinate which in this case is just -1
since 360 degrees is radians, what can we use that information for
we can use that information as a conversion factor to go between degrees and radians
how do we get the degrees of each quadrant
we get those degrees because since we are dealing with a circle we split that circle up into 4 quadrants which makes each quadrant 90 degrees and we just count from 0 -the degrees of each quadrant change based on whether or not we are going clockwise or counterclockwise -we can convert these from degrees to radians as well
how do we measure rotation
we measure rotation in either degrees or radians because they together are a conversion factor
how should we think of a circle
we should think of it as it costs 2π to draw a circle on a graph and that means that each quadrant is 0.52π of that circle because there are four quadrants -this is so that we can think of the unit circle in terms of
what do we use the unit circle for
when we make an angle on the graph, starting at standard position, the terminal side intersects the unit circle at a point on the graph and we can use sin and cosin to identify those (x,y) coordinates
terminal side
where the angle ends -it doesn't have to correspond to an axis it is just the side of the angle that isn't the initial side and thus where the angle ends
what is the pattern of a sine function which makes it a periodic function
x intercept, max, y intercept, miniumum, x intercept -this together is always one complete cycle
what is the cosine function
y = a cos(bx)
what is the sine function for graphing
y = a sin(b∅)
what is the cosine function parent graph
y = cos(x)
sine function
y = sin(∅) aka y = sin(x)
is the graph of y = sin (-x) and the graph of y = - sin (x) the same
yes because both have a negative -even though the negatives are in different spots, sine functions are odd which means that it extrapulates the negative from the function and applies it to it which is why both of those graphs will look the same and start below the x axis
will this graph start below or above the x axis (it will always initially start at the origin) y = sin (-x)
yes because it doesn't matter where the negative is for sine functions but rather that there is one and so even though the amp is positive, the negative b value makes it negative and is why it will start below the x axis -sine functions are odd which means that it extrapulates the negative from the function and applies it to it which is why both of those graphs will look the same and start below the x axis
do cycles go in both directions
yes!
standard position
you are going to start at the positive x axis -the vertex of this angle is the origin and the starting ray is the positive x axis -an angle is in this when its initial side lies along the x-axis and its endpoint is at the origin
how do you find the period of a cosine function y = a cos(bx)
you find the period the same way as you do with the sine function 2π / b
how do you find the subintervals of a cosine function y = a cos(bx)
you find the subintervals the same way you do for a sine function -Subs = period * ¼
how do you find the range of a cosine function
you have to use the amplitude -the range will always look like: negative amp ≤ y ≤ positive amp e.g. y = 3cos(4x) -the amplitude here is 4 so our range will be -3 ≤ y ≤ 3 -the reason this is our range is because the amplitude is the maximum and minimum our graph can go which means that all of the y values in the graph are either one of those two values or between those two values
how do you find the range of a sine function
you have to use the amplitude -the range will always look like: negative amp ≤ y ≤ positive amp e.g. y = 3sine(4x) -the amplitude here is 4 so our range will be -3 ≤ y ≤ 3 -the reason this is our range is because the amplitude is the maximum and minimum our graph can go which means that all of the y values in the graph are either one of those two values or between those two values
when given a sine curve on a graph how do you find b
you look for the last x value in the cycle and set that equal to the equation for the period and then cross multiply to find b
when given the amplitude and period how do you find b
you set the period equal to the equation for the period (2π/b) and then cross multiply to find b