trig exam 1
5 quarter points of sin
(0,0) (pi/2,1) (pi,0) (3pi/2,-1) (2pi,0)
5 quarter points of cos
(0,1) (pi/2,0) (pi,-1) (3pi/2,0) (2pi,1)
phase shift formula
C/B (this will be the x-coordinate point for the first quarter point)
Length of intercepted arc
S
Vertical shift of sin cos equation
Y=Asin(Bx-C)+D Y=Acos(Bx-C)+D D is the vertical shift
horizontal shift equation
Y=sin(X-C) Y=cos(X-C)
Unit Circle
a circle with a radius of 1, centered at the origin
radian
a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.
how to find coterminal angles
add or subtract 360 or 2π
cos theta (right triangle)
adj/hyp
Cot theta (right triangle)
adj/opp
quad 1
all trig functions are positive
general angle
angles not restricted in size that can be positive, negative, or zero
t<0
clockwise
quadrantan family of triangles
consist of angles in standard position that are coterminal with 0 (positive x axis) pie/2 (positive y axis) pie (negative x axis) 3pie/2 (negative y axis)
pie/3 family of triangles
consist of angles in standard position that are coterminal with pie/3 (quad 1) 2pie/3 (quad 2) 4pie/3 (quad 3) 5pie/3 (quad 4)
pie/4 family
consist of angles in standard position that are coterminal with pie/4 (quad 1) 3pie/4 (quad 2) 5pie/4 (quad 3) 7pie/4 (quad 4)
pie/6 family
consist of angles in standard position that are coterminal with pie/6 (quad 1) 5pie/6 (quad 2) 7pie/6 (quad 3) 11pie/6 (quad 4)
quad 4
cos and sec are positive
t>0
counterclockwise
cos characteristics
domain (-infinity, +infinity) range [-1,1] periodic function w/ period od 2pi cos x= cos(x+2pin) y intercept 1 x intercepts/zeros x=(2n+1)pi/2 even function/ symmetric about y axis cos(-x)=cos(x) relative max: x=2pin relative min: x=pi+2pin
characteristic of sine graph
domain (-infinity, +infinity) range [-1,1] periodic w/ period of 2 y intercept 0 x intercepts/ zeros x=npi odd function/ symmetric about orgin sin(-x)=-sinx relative max: x=pi/2 + 2pin relative min: x=3pi/2 + 2pin
x^2+y^2=r^2
equation of a circle
SEC theta (right triangle)
hyp/adj
CSC theta (right triangle)
hyp/opp
if C<0...
shift C units to the left
if C>0...
shift C units to the right
if D<0...
shift D units down
if D>0...
shift D units up
tan functions of unit circle
sin t=y cos t=x tan t=y/x , x cannot = 0 csc t= 1/y, y cannot = 0 sec t=1/x, x cannot = 0 cot t=x/y, y cannot = 0
Confunction Identities
sin theta=cos(pie/2-theta) cos theta=sin(pie/2-theta) tan theta=cot(pie/2-theta) sec theta=csc(pie/2-theta) csc theta=sec(pie/2-theta) cot theta=tan(pie/2-theta)
Pythagorean Identities
sin^2x+cos^2x=1 1+tan^2x=sec^2x 1+cot^2x=csc^2x
quad 2
sine and csc are positive
Reciprocal Identities
sinθ = 1/cscθ ; cscθ = 1/sinθ cosθ = 1/secθ ; secθ = 1/cosθ tanθ = 1/cotθ ; cotθ = 1/tanθ
quad 3
tan and cot are positive
Quotient Identities
tanθ = sinθ/cosθ cotθ = cosθ/sinθ
rotated side
terminal side
reference angle
the acute angle formed by the terminal side of an angle in standard position and the x-axis
fixed ray
the initial side
1/2pie radian
1 rad
steps for sketching functions in the form of y=Acos(Bx) and Y=Asin(Bx)
1. if B<0 use the even/odd properties to rewrite it as B>0 2. determine amp |A| and range [-|A|, |A|] 3. determine period p=2pi/B 4. Interval for one complete cycle [0,(2pi/B)] subdivide period by 4, start with zero and add ( (2pi/b) / 4) to the x-coordinate of each quarter point 5. multiply y quarter points by A
steps for sin cos phase shift
1. rewrite equation to y=Asin(Bx-C/B) and y=Acos(Bx-C/B). make sure B>0 2. determine amp |A| and range [-|A|, |A|] 3. determine period p=2pi/B 4. phase shift C/B 5. an interval for a complete cycle [C/B, C/B + P] C/B is my first x-coordinate subdivide P by 4 start with C/B and add P/4 to the x coordinate each quarter point 6. multiply y by A
steps for vertical shift
1. rewrite equation to y=Asin(Bx-C/B)+D and y=Acos(Bx-C/B)+D. make sure B>0 2. determine amp |A| and range [-|A|+D, |A|+D] 3. determine period p=2pi/B 4. phase shift C/B 5. an interval for a complete cycle [C/B, C/B + P] C/B is my first x-coordinate subdivide P by 4 start with C/B and add P/4 to the x coordinate each quarter point 6. multiply y by A then add D
determining the equation for y=Asin(Bx) and y=Acos(Bx)
1. the equation is Y=Asin(Bx) if the graph passes thru the orgin and Y=Acos(Bx) if it does not pass thru the orgin 2. determine P and use the equation P=2pi/B to determine B
(pie)radian
180 degrees
one complete revolution
2pie
2(pie)radian
360 degrees
complete counterclockwise
360 degrees
arc length S(unit circle)
=theta r
general angle of trig functions
if P(x,y) is a point on the term side of any angle in standard position and if r=sq root (x^2+y^2) is the distance from the orgin to point p
negative radians
look at notes
positive radians
look at notes
negative degree angle
look in notes
positive degree angles
look in notes
amplitude of a sin/ cos function
measure of half the distance between the min and max points y=Asin(Bx) y=Acos(Bx)
radians to degrees conversion
multiply by 180/pie
Degrees to radians conversion
multiply by π/180
if theta is positive and greater than 2pie
negative values of k when finding coterminous
Tan theta (right triangle)
opp/adj
sin theta (right triangle)
opp/hyp
period
p=2pi/B B>0
general angles sec theta=
r/x, x cannot be 0 (undefined)
general angles CSC theta=
r/y , y cannot be 0 (undefined)
distance formula for angles
r=sq root (x^2+y^2)
smallest angle greater than 0 degrees that is coterminal
theta C
(ref angle) if term side of theta c lies in quad 4...
theta R=2pie-theta C (R=360-C)
(ref angle) if term side of theta c lies in quad 2...
theta R=pie-theta C (r=180degrees-C)
(ref angle) if term side of theta c lies in quad 1...
theta R=theta C
(ref angle) if term side of theta c lies in quad 3...
theta R=theta C-pie (R=C-180)
coterminal angles
two angles in standard position that have the same terminal side
angle
two rays with a common endpoint
general angles cos theta=
x/r
general angles cot theta=
x/y, y cannot 0 (undefined)
general angles sin theta=
y/r
general angles tan theta=
y/x , x cannot be 0 (undefined)
