Unit 2

every point on the unit circle has the coordinates of

(cosθ, sinθ) because the terminal ray is the radius, which equals one. If you draw a right triangle from the endpoint of the terminal to the x axis, then the horizontal line is the x coordinate and the vertical line is the y coordinate. Sine of the angle is opposite/hypotenuse which is y/1, so sinθ is the y coordinate. Cosine of the angle is adjacent/hypotenuse which is x/1, so cosθ is the x coordinate.

tricks for previous slide

-if you have sec of something, and you move it around so that you are now applying something to sin as opposed to cosin, you will change it to csc(x) -always turn tan into sin/cos -cos²+sin²=1, 1-cos²=sin², 1-sin²=cos² -if you divide above equation by sin or cos, then you get equations with sec, tan and csc 1+cot²=csc² 1+tan²=sec² -sin*csc, cos*sec, tan*cot all =1

unit circle Radius: Circumference: Equation

1 2π x²+y²=1

360 degrees= ? radians 1 degree= 1 radian=

2π π/180 radians 180/π degrees (about 57°)

radian

Number of raduis's in the arc of the angle. angle measure that is equal to the arc length in radians from 0,0 to endpoint of terminal ray

reference angle

The acute angle formed by the terminal side of θ and the x-axis (they add to 180 degrees)

coterminal angles

angles that share a terminal side, and differ by a multiple of 360°; for example, 32° and 32°+360°=392° are coterminal

when sin(x+anything) or sin (-x) etc, just think about it

as if you are moving around the angle. remember to see where congruent triangles. Remember to look at the whole angle, not just the acute version of the angle. also it is helpful to pick a starting angle that has an obvious long and short side, because otherwise it can get confusing

when graphing sin/cosin functions with transformations,

as opposed to starting with the beginning equation and then doing transformations, it is easier to find Sinusoidal axis=d amplitude=a period=360°/b or 2π/b horizontal phase shift=c domain range a cos ( b ( x + c ) ) + d

graph of y=sec(x)

can check this with the main numerical points. also can think about it is starting with cos graph and flipping the loops up and down looks like this because it is dividing by cos, so there will be undefined moments etc

converting minutes and seconds into degree

degree + minutes/60 + seconds/3600

How to simplify really complicated fractions made up of different trig functions that have arguments that are not just the angle (eg cos(x-pi/2)

draw a unit circle and start with a random angle. then do the rotation that the section codes for (eg if cos(x-pi/2), then move the angle around pi/2. whatever the cos of that new angle is is what you turn the final thing into. so cos(x-pi/2)=sin(x)). And then you just simplify as far as you can get

how to find the four smallest positive solutions to trigfunction(x)=number

draw those angles on a unit circle. if it is a good number, you should know what the angle is based on the unit circle. from there, just keep adding 2pi or 360. (wait for future flashcard for a more complicated explanation)

when given a point that is on the terminal ray, you can find the trig functions by

drawing the larger triangle outside of the unit circle and then finding the hypotenuse by doing a²+b²=c² and then draw a small triangle inside the unit circle that is similar and find the lengths of the smaller triangle's sides by doing similar triangle ratios

if trying to see how far or how fast a wheel is turning,

find the circumference, and then use conversions to figure other stuff out

converting decimal degree to minutes and seconds

first convert decimal part to minutes by multiplying by 60, then convert the remaining decimal to seconds by multiplying it by 60

If there is an absolute value on a sin/cos function,

first just graph the function, and then do absolute value as tranformations

to find the solution to a sinusoidal equation or inequality like y=a cos ( b ( x + c ) ) + d, where you have every value but x,

first simplify as much as you can by subtraction and division until you get to cos(b(x+c))=stuff. Stuff will be something that you can draw on the unit circle like 1/2. so then you know that b(x+c)=options (if 1/2, then it would be π/3 or 5π/3), and then you can simplify from there. remember to include the domain restrictions. if it's any possibility, say +2πn with n=all integers domain: -if you are finding all solutions, you will say +2πn, with n being all integers -if the period changes (so you are multiplying the angle(the x part of the equation) by something) then you would divide 2πn by that multiplyer. If the only transformations on the angle is translation, then you don't change the 2πn. -another way to think about the domain besides just looking at how many times you go around the circle in a negative or positive direction, is thinking about the equation as a graph. you draw the graph, and then a horizontal line at what the graph equals. The intersection points are the solutions. You can then include the domain restriction, and the amount of periods behind and in front of the center represent how many times you are going around the circle, and the period length represents how much you add each time to represent one rotation

horizontal phase shift

how much is it translated horizontally Phase and vertical shift refer to the transfer of a function in the horizontal and vertical directions, respectively. For the function f(x)=Asin(Bx+C)+D, where A, B, C, and D are constants, the phase shift is defined as C/B. If the phase shift is positive, the shift occurs to the left; if the phase shift is negative, the shift occurs to the right.

when a word problem talks about day and night

it is a sinusoidal function

to do inequalities with trig functions,

just find the points on the unit circle that it would be if it was equal, and then shade in based on when either the x,y, or tan value is greater. if it is a<x<b, shade in between.

graph of y=csc(x)

like sec, but starting with SIN graph and flipping up and down

in a word problem where the circle is bigger than a unit circle, you can turn it into a unit circle by or you can find sin and cos as if it is a unit circle, and then multiply by the radius at the end (because if you just do sin, it is for a radius of 1, so you multiply it by whatever the new radius is)

looking at lengths as fractions. eg if the radius is 5, and a ray goes from the center and intersects with the circle at (4,3) then cos x=4/5 and sinx=3/5

graph of y=cot(x)

makes sense because you are changing where the vertical asymptote is

in word problem, you can find number of revolutions by finding:

number of meters/(meters per revolution. which is the circumference) then you will get a number with a decimal. lets say 11.45. so there are 11 revolutions, plus .46 of a revolution, so .46 times 2pi, and that gets your angle

remember! cos and sin cannot be greater than

one, because that is the radius of the unit circle

graph of y=tan(x)

remember this shape, but you can also think about it numerically, finding the intersection points or vertical asymptotes. also, it is like a rational function because tan is a fraction, it gets really tall really fast because you are dividing larger numbers by smaller numbers, the vertical asymptote is where you are dividing by zero

when it is sin(pi-x)

rotate pi and then go back x

sine, cosine, tangent, cotangent, secant, cosecant

sin reciprocal of csc cos reciprocal of sec tan reciprocal of cot sin and cos complementary tan and cotan complimentary sec and csc complimentary

if given one trig function of an angle (eg sinx=1/5), you can find the others by

sin²x+cos²x=1 plugging in and then doing reciprocals

amplitude

the distance from the centerline to the minimum or maximum values (so this will always be positive because it is the distance)

sinusoidal axis

the horizontal centerline

period

the horizontal distance before the graph repeats itself

how to think of graphing y=sin(x) or y=cos(x)

the x (the input) is the angle on the unit circle in radians, and then the y (the output) is either the x or y coordinate depending if sin or cos

standard position of unit circle

vertex is at origin, and initial ray is on x axis

if sin(x)=sin(y),

x=y

easy way to write an equation when given a sinusoidal graph

y=a cos ( b ( x + c ) ) + d first of all, it's easier to have b outside the parenthesis because then you are first doing the period change and then moving over for a, find the amplitude by looking at distance from centerline to height. for d, find centerline. for b look at the period (360°/b=new period), and for c look at horizontal translation (make sure to pay attention to if sin or cos

when cos(x)^2>something, or when cos^2>sin^2

you can square root both sides as long as you think of it as distance of x and y as opposed to negative and positive you can also look at it like an equation and find the points of intersection and then see where it is greater than or less than