Graphing tangent and reciprocal

Pataasin ang iyong marka sa homework at exams ngayon gamit ang Quizwiz!

Take the _____ of A

Absolute value

Finding your bearings

Always find them by looking at north or south line first

In cosine, _____ equals 0 In cosine, the reference angle is the _____ of the other angle Which numbers are neg and pos?

Anything over 2 that doesn't reduce to whole number like 5pi/2 or 3pi/2 Opposite sign! For example pi/6 is sqrt3/2 but 5pi/6 is -sqrt3/2

Find the exact value of each of the remaining trig functions of theta. Cos theta=-24/25, theta is in quad 3

Draw out triangle Y is negative because theta is in quad 3

Composite functions! f(f-1(x))=x. f-1(f(x))=x

F-1(x) must be in domain of f f(x) must be in domain of f-1

Graphing tangent functions

Find consecutive asymptotes using -pi/2<bc-c<pi/2 Find midway between asymptotes Find 1/4 pt ( LA plus mdpt divided by 2) and 3/4 pt (mdpt plus RA divided by 2) Y-copies -A and A respectively

Sin ( sqrt2/2) vs sin-1(sqrt2/2)

First gives decimal second gives pi

INVERSE PROPERTIES tan(tan-1x)=x tan-1(tanx)=x

For all reals For all x in (-pi/2, pi/2)

KEEP IN MIND: If there's a negative before the function...ex: y=-2cos2x LABEL POINTS SO MR. HAYBURN CAN SEE

If reflects over x-axis

The whole even and odd function rule doesn't hold true for...

Inverses Cos-1(-sqrt3/2) equals 5pi/6 not neg

Positive cosine graph look like

Maximum at origin

In sin cos and tan, the reference angles are all

Positive

Given sin theta=0.8836

Put calc in degree and use inverse button

If the absolute value of A is smaller than one, the graph...

Shrinks

When given sin(sin-1) or sin-1(sin) use

The inverse properties

If given sin (sin -1 pi) what happens

not in domain so its undefined when its the inverse in the parenthesis and its not in domain, its UNDEFINED

If given sin -1 (sin 5pi/4) what happens

number not in domain but since inverse is outside of parenthesis, you can still do it find sin 5pi/4 then then sin -1 of that

Pythagorean identities

sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) cot2(t) + 1 = csc2(t)

Y=sin-1x means ____ where

x=siny where -pi/2 is less than or equal to y which is less than or equal to pi/2 and -1 is less than or equal to x which is less than equal to 1

Sine inverse chart -1, -sqrt3/2, -sqrt2/2, -1/2, 0, 1/2, sqrt2/2, sqrt3/2, 1

-pi/2 -pi/3 -pi/4 -pi/6 0 pi/6 pi/4 pi/3 pi/2

Y=sinx has inverse on

-pi/2 lessthan or equal to x less than or equal to pi/2

Sin 0 Cos 0 Tan 0

0 1 0

Sin-1 0 Cos -1 0 Tan -1 0

0 Pi/2 0

Y=cosx has inverse on

0 is less than or equal to x less than or equal to pi

Cosθ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi Remember cosine pattern!

1 sqrt3/2 sqrt2/2 1/2 0 -1/2 -sqrt3/2 -1 -sqrt3/2 -1/2 0 1/2 sqrt3/2 1

Cosine inverse chart 1, sqrt3/2, sqrt2/2, 1/2, 0, -1/2, -sqrt2/2, -sqrt3/2, -1

1, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, pi

Number line

1/8 1/4 1/2 3/4 1 but with pi!

For example, y=4sin(2x-2pi/3) What's A B and C?

A is 4 B is 2 and C is 2pi/3

If given tan -1 (-1)

A quick way would be to take the negative out and get the answer then throw it back in

y=cotx domain and range

All real numbers except multiples of pi All real numbers (-inf, inf)

When putting sin(0.2) in calculator,

BE IN RADIAN MODE

Finding angle of elevation, calc should be in

Be in degrees

phase shift

C/B and it's horizontal

ON TEST: sin-1(1.3)

Can't do it because it's outside domain

Even functions

Cos sec

Find two values of theta, 0 less than equal to theta less Han equal o 2pi thst satisfy he equation: cos theta=sqrt2/2

Find where cos is positive then find two angles in those quadrants

Graphing tan in calc:

Put function in Y1 and put Xmin as -pi/2 and Xmax -3pi/2 and Xscale pi/4

ASTC?

Q1 all positive Q2 sin and Csc are + rest - Q3 tan and cot are + rest - Q4 cos sec are + rest are -

The domain of the restricted function for sinx cosx and tanx are the _____ for sin-1x cos-1x and tan-1x

RANGE

REMEMBER SIN AND COS PATTERN

Sin: 0 1 0 -1 0 1 0 -1 0 1

Amplitude changes...

The minimum and maximum NOT THE PERIOD

Function domain and interval The restrictions are...

The range for y=sin-1x y=cos-1x and y=tan-1x

How to solve sin49 csc49

They're reciprocals so sin49(1/sin49) Which equals 1

How to figure out if it's a function?

Vertical line test

Trig using y r and x Sin Cos Tan Csc Sec Cot

Y/r X/r Y/x R/y R/x X/y

y=cscx domain and range

all real except multiples of pi (-infinity, -1] U [1, infinity)

y=secx domain and range

all real numbers except odd multiples of pi/2 (-inf, -1] U [1, inf)

Finding the side length of triangle, put calc in

degrees

INVERSE PROPERTIES Sin(sin-1(x))=x Sin-1(sin(x))=x

for all x in [-1, 1] for all x in [-pi/2, pi/2]

Inverse and intercepts are

not the same

Where's tangent undefined??

pi/2 3pi/2

What's 30 45 60 and 90 in radians

pi/6 pi/4 pi/3 pi/2

put calc in degree or radians: cos -1 6/7

radians

Reciprocal Identities

sinθ = 1/cscθ ; cscθ = 1/sinθ cosθ = 1/secθ ; secθ = 1/cosθ tanθ = 1/cotθ ; sin/cos cotθ = 1/tanθ cos/sin

45-45-90 triangle

x, x, x√2

30-60-90 triangle

x, x√3, 2x x is across from 30 x√3 is across from 60 2x is across from the hypotenuse

What does y=sin-1x mean

x=siny

sin 0, cos 0, tan 0

0 1 0

Sin θ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi

0 1/2 sqrt2/2 sqrt3/2 1 sqrt3/2 1/2 0 -1/2 - sqrt3/2 -1 -sqrt3/2 -1/2 0 Remember sin pattern!

Tanθ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi

0 sqrt3/3 1 sqrt3 undefined -sqrt3 -sqrt3/3 undefined sqrt3/3 sqrt3 undefined -sqrt3 -sqrt3/3 0

GRAPHING VARIATIONS OF Y=SINX STEPS

1. Identify the amplitude and period 2. Find the x-value of the five key points (3 intercepts, 1 min, 1max) 3. Find the y-values 4. Connect points in a smooth curve 5. Extend graph as desired

In graphing tangent, how do you find 1/4 point? 3/4 point?

Add left asymptote and midpoint and divide by 2...use -A for y Add midpoint and right asymptote...use A for y

Odd functions

All functions are odd except cos and sec

y=cscx domain and range

All real except multiples of pi (-inf, -1] U [1, inf)

Y=tanx domain and range

All real numbers All real except odd multiples of pi/2

Y=cos-1x means

Cosy=x where 0 which is less than or equal y which is less than or equal to pi and -1 is less than or equal to x which is less than or equal to 1

Finding distance across a lake in which would be solving for one of the sides, make sure your calc is in

Degrees

Graphing a tangent function:

Do -pi/2<bx-c<pi/2 Find midpoint between them Then find 1/4 and 3/4 point

To find inverse sin cos and tan, just

Do the unit circle and draw a triangle using 45 45 90 and 30 60 90

Domain and range of tangent function?

Domain all real except multiples of pi/2 Range is all reals

Domain and range of y=cosx

Domain is (-inf, inf) Range is [-1, 1]

Domain and range of y=sinc

Domain is (-inf, inf) Range is [-1, 1]

Domain and range of y=cotx

Domain is all real except multiples of pi Range is all real

Domain and range of y=secx

Domain is all real except odd multiples of pi/2 Range is (-inf, -1] U [1, inf)

Domain and range of y=cscx

Domain is all real numbers except multiples of pi Range is (-inf, -1] U [1, inf)

The point given below is on the terminal side of an angle theta. Find the exact value of each of the six trig functions of theta. (3, -3)

Find r R=sqrt x2+y2 Then use the ratios for example sin is y/r

INVERSE PROPERTIES cos(cos-1x)=x cos-1(cosx)=x

For all x in [-1,1] For all x in [0, pi]

What happens if a negative is in front of sin or cos?

For sin when it's positive the pattern is 0 1 0 -1 0 When negative it's 0 -1 0 1 0 and reflect across x axis

Increment is

How far each point is from one another

In sine, the fractions that have a smaller number in the numerator, The fractions that have a larger number in numerator equal In sin, any number ___ is 0

It's positive Example: 2pi/3 and 5pi/6 They have the reference angle of pi/3 and pi/6 and equal sqrt3/2 and 1/2 Negative Any number like 2pi 3pi 11pi

For problems like sin(sin-1 ...)

Just cancel them out and use x When it written w positive sin then neg sine, it's just the number This goes for ALL trig functions

Graphing trig functions in calc...

Make sure to be in RADIAN mode

A negative sin graph looked like

Maximum then 0 then minimum

Negative cosine graph look like

Minimum at origin

A positive sin graph looks like

Minimum second quadrant then goes thru 0 then maximum in first quad

Does y=sinx pass the horizontal line test?

No, so we look at domain -pi/2< or equal to x< or equal to pi/2 which PASSES the HL test

Which trig functions have amplitudes

ONLY sin and cos

Is sin an even or odd function

Odd

Inverse function

Passes the horizontal line test If point (a,b) is in f(x) then point (b,a) is in f-1(x) It is the reflection of the function about the line y=x

cos-1(cos pi/2)

Pi/2 is in the domain so it's pi/2

The whole thing with inverse and no inverse is basically seeing what you're given and basically finding the reference angle...so Q1 Q2 Q3 Q4 When it's a non acute angle

Q1, it's just itself Q2 180- theta Q3 theta - 180 Q4 360 - theta 360 - theta And if it's negative, go counterclockwise

θ 0 30 45 60 90 0 π/6 π/4 π/3 π/2 SIN COS TAN

SIN 0 1/2 sqrt2/2 sqrt3/2 1 COS 1 sqrt3/2 sqrt2/2 1/2 0 TAN 0 sqrt3/3 1 sqrt3 undefined tangent is sin/cos so you can always get tangent

Which functions do you graph the same way?

SIN AND COS

Simplifying expressions of sin -1x involving algebraic expressions cos(sin -1 x)

Set sin -1x= θ sin θ=x which is sin θ=x/1 which is o/h So set up triangle like before on graph and find a Then evaluate it at cos Should be like sqrt 1-x^2

If given one trig function then another, example, cos(tan-1 tan 5/12) you...

Set the one inside the parenthesis to theta: tan-1 5/12= θ and then rewrite it like tan θ=5/12 Well tan is o/a which is y/x so set it up lie a triangle on coordinate plane! Then once you have that you evaluate it at the trig function in front of parenthesis! Which is cosine

Cofunctions

Sin and cosine are cofunctions: sin(pi/2-x)=cosx cos(pi/2-x)=sinx Tangent and cotangent are cofunctions: tan(pi/2-x)=cotx cot(pi/2-x)=tanx Secant and cosecant are cofunctions: sec(pi/2-x)=cscx csc(pi/2-x)=secx

Diff between sin of a regular fraction like sin sqrt2/2 or sin 1/2 and sin of a pi like sin pi/4

Sin of a fraction like sin 1/2 gives you just a decimal Sin of a pi like sin pi/4 gives you a fraction like the quiz we had

cos(cos-1(pi/2))

So pi/2 is within the domain so the answer is pi/2

Sin pi/4 Cos pi/4 Tan pi/4 Sin pi/3 Cos pi/3 Sin pi/6 Cos pi/6

Sqrt2/2 Sqrt2/2 1 Sqrt3/2 1/2 1/2 Sqrt3/2

What's the starting point?

Starting point is 0 when there's no c When there's c the stating point is C/B

If the absolute value of A is larger than one, the graph...

Stretches

Which trig functions are which period

Tan and cot have the period of pi or pi/b All the rest have the period of 2pi or 2pi/b

Questions about values and inverse's?

Use ASTC ALL STUDENTS TAKE CALC

Can't graph y= Csc sec or cot so you

Use reciprocal!

Graph Csc and sec?

Use reciprocals!! For Csc use sin For sec use cos The x intercepts of sin and cos are the VERTICAL ASYMPTOTES The minimum of sin and cos are the maximums The maximum of sin and cos are the minimums SKETCH!

Where do you put vertical asymptotes at on a tangent graph?

Vertical asymptotes at pi/2 and all odd multiples of pi/2(1 3 5 7)

Y=tanx vs y=tan -1 x

With the inverse, the x and y switch Domain of regular is (-pi/2, pi/2) and range is (-inf, inf) Domain of inverse is the range of regular, (-inf, inf) the range is the domain of regular (-pi/2, pi/2) Regular has a Points at (-pi/4, -1) (0,0) and (pi/4,1) Inverse has points at -1, -pi/4 and 0,0 and 1, pi/4 Regulars two vertical asymptotes of - and + pi/2 are now horizontal asymptotes

Graphing cot?

You'd think you'd use a reciprocal but graph it like tangent! Except use 0<bx-c<pi Find midpoint between them Find 1/4 and 3/4 Points THIS TIME..use A first then -A Think that the A's are switched because tan and cot are reciprocals

When finding SinA, you have to make sure

Your calculator is in DEGREE MODEEE

Function: y=tanx Domain of restricted function Interval on which the function complete one period

[-pi/2, pi/2] (-pi/2, pi/2)

Function: y=sinx Domain of restricted function Interval on which the function complete one period

[-pi/2, pi/2] [0, 2pi]

For finding the inverse, go by these restrictions ...KEEP THESE IN MIND EACH TIME YOU DO INVERSE Sin-1x Cos-1x Tan-1x

[-pi/2, pi/2] [0, pi] (-pi/2, pi/2]

Function: y=cosx Domain of restricted function Interval on which the function complete one period

[0, pi] [0, 2pi]


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