Chapter 7 test algebra
reference angle quadrant 2
180- theta or pi- theta
pythagorean triples
3-4-5 5-12-13 7-24-25 8-15-17 9-40-41
pi/6
30 degrees
unit circle: angles corresponding with 30 degrees and their radian measures
30, 150, 210, 330 pi/6, 5pi/6, 7pi/6, 11pi/6
reference angle quadrant 4
360 - theta or 2pi - theta
pi/4
45 degrees
unit circle: angles corresponding with 45 degrees and their radian measures
45, 135, 225, 315 pi/4, 3pi/4, 5pi/4, 7pi/4
pi/3
60 degrees
unit circle: angles corresponding with 60 degrees and their radian measures
60, 120, 240, 300 pi/3, 2pi/3, 4pi/3, 5pi/3
quadrant order
I- both positive II- x negative, y positive III- both negative IV- x positive, y negative
how to do 30-60-90s and 45-45-90
draw the triangle and find the trig functions for each one
angle of depression
falls from the horizontal NOT ALWAYS IN TRIANGLE ITS FROM THE HORIZONTAL!!! ex: building higher up
how to find trig functions through definition
find all the sides and then use soh-cah-toa
initial side
fixed ray the angle is generated from (on the x axis)
periodic function definition
functions that repeat at regular intervals
negative angles
go clockwise
positive angles
go counterclockwise
positive x axis
to the right where the numbers are positive
how to find an angle with trig functions
use the inverse functions in degrees
how to find trig functions on the calculator
use the sin, cos, and tan buttons. for the cofunction identities, you'll just do one divided by the cofunction. MAKE SURE YOURE IN THE RIGHT MODE!!! if there is no degree symbol, its radians.
how to find trig functions with identities
you can't draw a triangle and you have to use all the different identities you know to find it
reduced triangle principle
you could reduce all the sides of the triangle by a common multiple to make it easier for trig functions
what to do to find two sides in word problem
you use the angles and the side and trig functions. but when you find the second one DONT use the new side you find because its an approximation
signs for quadrants
"all students take calculus" Quadrant 1: all are positive Quadrant 2: sin and csc are positive Quadrant 3: tan and cot are positive Quadrant 4: cos and sec are positive
coordinate points 180 degrees
(-1,0)
coordinate points 270 degrees
(0,-1)
coordinate points 90 degrees
(0,1)
coordinate points 0/360 degrees
(1,0)
unit circle 60 coordinate points
(1/2, root 3/2)
how to find trig functions from the unit circle
(cos, sin)
unit circle 45 coordinate points
(root 2/2, root 2/2)
unit circle 30 coordinate points
(root 3/2, 1/2)
when can you use (cos, sin)
-on the unit circle -quadrantal angles -when the triangle's hyp = 1 -Q1: special 30, 60, and 45s
sin (-θ) / csc (-θ)
-sinθ -cscθ
tan (-θ) / cot (-θ)
-tan θ -cot θ
unit circle: quadrantal angle measures and their radian measures
0, 90, 180, 270, 360 0, pi/2, pi, 3pi/2, 2pi
how to find the trig functions of an angle on a coordinate plane (not necessarily on the unit circle)
1) draw the terminal side/plot the point it gives you 2) drop an altitude onto the x axis 3)use the x and y values to find the legs, then either use triples or the pytag theorem 4) then use the reference angles for the values and ASTC for signs
how to draw angles on coordinate plane
1) put the initial side on the positive x axis 2) note that the axises are 90, 180, 270, 360/0 degrees and pi/2, pi, 3pi/2, 0/2pi radians 3) put the terminal side in the quadrant it belongs 4) draw arrow to show which way the angle is rotating
how to use reference angles to find trig functions
1. determine the quadrant of terminal side angle 2. find reference angle 3. evaluate trig function of reference angle, decide if positive or negative from "all students take calculus"
area of a sector
1/2 times r^2 times central angle IN RADIANS
cos
adj/hyp
standard position
an angle with the vertex at the origin and initial side on positive x axis
coterminal angles
angles in standard position with the same terminal side (infinite amount)
how to find coterminal angles
by adding and subtracting multiples of 360 or 2pi. might have to do it a couple times to get it positive or negative. you gotta make sure its in the parameters of what it wants it in. also to state what quadrant it is in, you can draw it to figure out its location. if it lies on an axis, you gotta say positive or negative as well
complementary angle theorem
cofunctions of complementary angles are equal. basically if you have a function of an angle, take the cofunction (ex: sin goes to cosine) and then its complement in either degrees or radian
periodic property: cos/sec
cos (θ+2π) = cos θ same w/ sec -also works w/ multiples of 2π and 360 degrees
even trigonometric functions
cos, sec
cos(-θ) / sec (-θ)
cosθ secθ
cotangent quotient identity
cot=cos/sin
reciprocal identities
csc= 1/sin sec= 1/cos cot= 1/tan
arc length
labeled as S S= radius times central angle IN RADIANS
how to find values w/ comp angle theorem and identities
make sure the degree measures are the same first! (WITH COFUNCTIONS sin/cos, csc/sec, tan/cot) rewrite things until you can visually cross things out or subtract the same thing SHOW ALL WORK. if its subtraction, look at pythagorean identities. if multiplication, quotient
how to convert radians to degrees
multiply by 180/pi
how to convert degrees to radians
multiply by pi/180
quadrantal angles
on an axis. past 360 degrees
tan
opp/adj
sin
opp/hyp
characteristics of unit circle
radius = 1 center = origin
sec
reciprocal of cos
csc
reciprocal of sin
cot
reciprocal of tan
angle of elevation
rise from horizontal ex: ground
terminal side
rotated ray that determines measure of angle
reference angle quadrant 1
same as theta
periodic property: sin/csc
sin (θ+2π)= sinθ same w/ csc -also works w/ multiples of 2π and 360 degrees
odd trigonometric functions
sin, csc, tan, cot
cofunctions for complementary angle theorem
sin/cos csc/sec tan/cot
pythagorean identities
sin^2 theta + cos^2 theta =1 tan^2 theta + 1 = sec^2 theta cot^2 theta + 1 = csc^2 theta
periodic property: tan/cot
tan (θ+π)= tanθ same w/ cot -also works w/ multiples of π and 180 degrees
tangent quotient identity
tan= sin/cos
reference angle definition
the acute, positive angle formed by the terminal side of theta and the x axis
reference angle quadrant 3
theta - 180 or theta - pi