trig exam 1

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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)


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