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