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