unit 3 precalc

how to find x in (0,pi) for cos(x/2)=cot(x)

(0,pi) is interval, so adjust the window X value. imput both equations into y= have to put (tan(x/2))-1 rather than tan-1(x/2) then solve and find intersection point

write 2pi/3 in degrees

(2pi/3)(180/pi) pi's cancel, left with 180(2/3) 120

angle of depression

---------------- \x \ \

law of sines ______ ambiguous cases, and law of cosines _______ ambiguous cases

1. has, meaning it can have 0,1,2 solutions 2. does not have, meaning it only ever has 1 solution

sin 30, 60, 45

1/2, rad3/2, 1/rad2 (rationalized- rad2/2)

tan 30, 60, 45

1/rad3, rad3/1, 1/1

write 135 in radians

135(pi/180) simplify to 3pi/4

triangles have a total of _____ degrees

180

to convert radian measure to degrees, multiply by

180/pi

what is 360 in radians

2pi

one degree = ______ minutes

60

one minute = _____ seconds

60

area of a sector

A=1/2r^2(theta) Theta is in RADIANS

area of triangle

Area= √s(s−a)(s−b)(s−c). s= (perimeter of triangle)/2

angle of depression facts

The angle of depression is always OUTSIDE the triangle. It is never inside the triangle. It is a downward angle from a horizontal line. the horizontal line in the angle of depression diagram is PARALLEL to the ground level.

angle of elevation facts

The angle of elevation is always measured from the ground up. It is an upward angle from a horizontal line. It is always inside the triangle

y=secx

VA at every zero of y=cosx

y=tanx

VA- cosx zeros, first at pi/r, then 3pi/2 {x|x=pi/2 +kpi, kEz}

y=cscx

VAs at zeros of sin

angle of elevation

\ \ \ \ x \ --------

cosine

adj/hyp

cotangent

adj/opp

the hypotinuse is

always positive

reference angles

always positive and always acute

cos-1/sin-1(ratio)

angle

sin/cos/tan ^-1 (ratio) =

angle

Law of Cosines

a²=b²+c²-2bcCosA b²=a²+c²-2acCosB c²=a²+b²-2abCosC

oscillation of sine and cosine graphs (this is a part of graph analysis)

basically the range, parent function is -1 and 1

decimal degree form

change DMS form back to a degree

steps to solve: find the exact vlaue of cos-1(-rad3/2)

cos-1(ratio)= angle finding the angle!!! find the triangle that would provide that ratio find the reference angle after that

x value

cosine

1/tan

cot

1/sin

csc

sine and cosine parent function domain and range

d- (-infinity,infinity) r- [-1,1]

when solving things like sin(33)=x, what mode on the calculator should you be in

degree

law of sines solved in which mode

degree mode

DMS form and notation

degrees (o), minutes ('), seconds (")

rationalizing trig ratios

don't do it

angle of depression and the angle of elevation are

equivalent

"solve the triangle" means

find all angles and sides

how to solve angle of depression/elevation

find its compliment or find the angle of elevation, calc should be in degree mode

how to solve- exact value of arcsin(-rad3/2)

find on unit circle where the ordered pair has sin value of -rad3/2 (which would be the y value), and figure out the corresponding reference angle, ANSWER IN RADIANS

how to put degrees into DMS form

given: 329.125 degrees 329^o + 0.125(60'/1^o) 329^o + 7.5' 329^o + 7' + 0.5(60"/1') 329^o 7' 30"

how to put DMS form into decimal degree form

given: 35^o 12' 7" 35^o + 12'(1^o/60') + 7"(1'/60")(1^o/60') 35^o + 12/60 + 7/60^2 35.202^o

csc and sec graphs

graph sin and cos with the transformations, remember the period is 2pi, then draw on the asymptotes (wherever theres a zero) and then draw the csc and sec graphs

secant

hyp/adj

cosecant

hyp/opp

identifying all angles coterminal with something

if you are given a trig value, remember all the possibilities to get that value given the trig values. for example, if you are given csc=-2, there are two possible ways you can get that value, using the unit circle, both negative. meaning you would have to have two separate coterminal equations.

how can two solutions occur

if you get a given angle, say 45, and you solve the sine and get an angle, say 70, you can add 45 and 70 (115) and subtract that from 180 to get the final angle (65). another method would be subtracting 70 from 180 (110), which would be the first angle, adding the given 45 and 110 together to find the final angle (25) IF EITHER SET OF ANGLES DON'T ADD UP TO 180- ONE SOLUTION

arcsin/cos/tan

inverse

why do you have to restrict inverse functions

it has to pass the VLT

"find the exact value of" sin4pi/3

just means find the refrence angle, determine if its 306090 or 454590 triangle, find respetive sides, then find the ratio

solve a triangle given two sides and an angle

law of cosine first to find the last side- c2 = a2 + b2 − 2ab cos(C) then use the law of sines to find the smaller of the other two angles- a/sin A = b/sin B = c/sin C then add up the triangles' angles and sub from 180 to find the final angle

how to find a side given an angle and two sides (SAS)

law of cosine- c2 = a2 + b2 − 2ab cos(C)

end behavior of sine and cosine graphs

limits do not exist

terminal side

line drawn off of x axis

find exact value of sin^-1(4/9), if it exists-- what is this asking

means what angle would one have to take the sin of in order to get 4/9, find the angle

sine x intercept

n(pi), nEZ

cosine x intercept

n(pi/2), nEZ

clockwise

negative

oblique triangles

not 90 degrees

initial side

on x axis

tangent

opp/adj, also sin/cos-- slope

sine

opp/hyp

cot and tan graphs

period of one, so when dividing by b to get period, divide pi/b tan- half graph negative, half graph positive, asymptotes at pi/2 for parent function, positive slope from left to right cot- entire graph positive, negative slope from left to right so shifts and such the same way as cos/sin (except period) shifts affect asymptotes too

what is 180 in radians

pi

to convert a degree measure to radians, multiply by

pi/180

counterclockwise

positive

reference angle rules/how to solve for reference angle

quadrant one: theta quadrant two: 180 (or pi) - theta quadrant three: theta - 180 (or pi) quadrant four: 360 (or 2pi) - theta

cos 30, 60, 45

rad3/2, 1/2, 1/rad2 (rationalized- rad2/2)

calculator mode when graphing

radian

what units do you include in problems without specified units

radians

cos/sin(angle)=

ratio

sin/cos/tan(angle) =

ratio

central angle is the same as the

reference angle

arc length

s=rtheta where r= radius , theta= central/reference angle

1/cos

sec

law of sines

sinA/a=sinB/b=sinC/c

y value

sine

sine starts up or down cosine starts up or down

sine starts down at origin cosine starts up at y=1

sine function is even or odd cosine function is even or odd

sine- even, symmetric to origin cosine- odd, symmetric to y axis

easy way to graph arcsin/cos/tan

switch x and y, physically turn paper, graph as a regular, non-inverse graph, then restrict it

coterminal angles

two angles in standard position that have the same terminal side

steps to finding the area of a sector

use a=(1/2)(r^2)(theta) 1. r^2 2. r^2/2 3. (r^2/2)(theta) just mult the coefficients 4. simplify

given ordered pair find six trig functions

x value is the initial side y value is the other side, terminal side found by Pythagorean theorem

equation of sine and cosine graphs

y= a sin/cos (bx+c)+d

what does each variable indicate in the equation if sine and cosine graphs

y= a sin/cos (bx+c)+d amplitude- |a| period- 2pi/|b| frequency- |b|/2pi or 1/period phase shift- -c/|b| vertical shift- d midline- y=d

also review every previous quizlet made for this unit as well as previous units (review questions are 10% of the test)

yikes

coterminal angle notation, including one positive and one negative angle

{theta| angle + 360 (or 2pi) k, kEz} angle + 360= positive angle - 360= negative