Graphing using angle of inclination, period, and amplitude

This past week we’ve learned about an alpha (angle of inclination) period, and amplitude.

-m=tan(alpha) where m is the slope of a line. “alpha” is also known as “angle of inclination.”

-tan2d= B/A-C à finds the angle of inclination for a conic by using AX^2 + BXY +CY^2 + …

Amplitude = how high/low a curve is from the origin. “A” is the variable used.

-To find amplitude:

Y=AsinBx

Y=AcosBx

*Amplitude cant be negative.

A negative in front of sin/cos flips the curve.

Period= how long it takes for a curve to repeat itself.

-To find period use the formula: 2(pi)/B

Steps to graphing a trig function:

1. Identify if theres a negative and which way it will start.
2. Find the period
3. Find the amplitude
4. Write your 5 important points (0, pi/2, pi, 3(pi)/2, 2(pi))
5. Divide every point by B
6. If anything is added/subtracted in parenthesis do the opposite for all 5 points
7. Sketch
8. Shift points up or down if anything is added/subtracted outside of parenthesis.

*It goes the same direction.

GRAPHING THIS SQUIGGLY LINE

• Identify if there is a negative and which way it will start
• Find the period by using the formula 2pie/b
• Find the amplitude (the x's coefficient)
• Write the 5 points (o, pie/2, pie, 3pie/2, 2pie)
• Divide every term by the "B" term in your given equation
• If anything is added/subtracted then, do the opposite to get 5 new points
• Sketch
• Shift points up or down if, anything is added or subtracted outside of the parenthesis

Example: -2Sin (pie/2x-1)

1) Sketch to get a visual

2) Find period using formula. P=4

3) Amplitude=2

4) Add 1 to your 5 points. New points: 1,2,3,4,5

5) Sketch

Solving Trig Functions!

Steps to solving trig functions:

~First you isolate the trig function.

~Second you take the inverse of the trig function

~Third you use the trig chart or a calculator to find the answer

~Fourth you use the quadrants to find the right angle

inverse would

~You must have at least two answers for each problem.

_______________________________________________________________

Ex.1: 4siny+5=3

~First subtract 5 on each side: 4siny=-2

-Next divide each side by 4: siny=-2/4:-1/2

~The inverse will be y=sin^-1(-1/2)

~sin^-1(-1/2)=30

~Now you use the quadrants to find your answers.

-sin is negative in quadrants 3 and 4. 30+180=210; -30+360=330.

~Your two answers will be 210 and 330.

Ex.2: 3cos(theta)+3=8

~First subtract 3 on each side: 3cos(theta)=5

-Next divede each side by 3: cos(theta)=5/3

~The inverse will be (theta)=cos^-1(5/3)

~cos^-1(5/3)=undefined

~You cant use the quadrants to find your answers because your answers are undefined.

trig inverses

In class this week we learned about solving trig functions.

Steps to solving:

First you have to isolate the trig function.

Next,you take the trig functions inverse. (-1)

Inverses are used to find angles.

You can either use your trig chart or calculator to find the answer.

The quadrants will help you determine whether if the angle is negative or positive.





1.) solve for theta(q)

6cos q + 9 = 7

subtract 9 from both sides

so then you have 2cos q = -2

divide both sides by 2

cos q = -2/6

q=cos^-1 (-2/6)= 109.471(since cos is negative in quadrant 3 and not 2 you cant use this.)

to move to quadrant two from one you have to make negative and add 180 which =70.529

to get your second answer take 70.529 and add 180 to get your third quadrant answer, which = 250.529

q=70.529 and 250.529,you then have to convert it to degrees,minutes,and seconds.

.529*60 = 31.74

.74 * 60 = 44.4

= 70 degrees 31 minutes and 44 seconds

.529* 60 = 31.74

.74 * 60 = 44.4

= 250 degrees 31 minutes and 44 seconds

q = 70 degrees 31 minutes and 44 seconds and 250 degrees 31 minutes and 44 seconds

Graphing Trig Functions

1.) Check if there's a negative and which way the graph will start.

2.) Find the period. [P=2pi/b]

3.) Find the amplitude(how many points up and down the graph goes).

4.) Write the five important points(0, pi/2, pi, 3pi/2, 2pi)

5.) Divide all points by B.

6.) If something is added or subtracted within parentheses, do the opposite on all five points.

7.) Sketch the graph.

8.) Shift the points up or down if anything was added or subtracted outside of parentheses.

_______________________________________________________________________

Example:

y=2sin 2x

A=2, B=2

1.) Sketch. Looks all squiggly. See figure A.

Figure A. (~~~~~~~)

2.) P=2pi/b => P=2pi2 => P=pi

3.) Amp=2

4.) 5.) 6.)

0 = 0/2 = [0]

pi/2/2 = [pi/4]

pi/2 = [pi/2]

3pi/2/2 = [3pi/4]

2pi = [pi]

7.) Sketch:

0 pi/4 pi/2 3pi/4 pi

~~~~~~~~~~~~~~~~~~~~~~~~~

8.) Doesn't apply.

8-1

M=tan(alpha). The slope of the line is m. Alpha is also known as angle of inclination.

Tan 2(theta)= b/a-c which finds the alpha of a conic ax^2+bxy+cy^2...=

Solve

Ex. 1. 4 sin theta=3
First divide both sides by 4
Theta= sin^-1 3/4 which is 48.590 degrees
Sin is positive so it's in the first and second quadrant. To move to the second quadrant make it negative and add 180 which gives you 131.41. You must then convert them to degrees minutes seconds. So to put 48.590 in that form first multiply .590 by 60 which gives you 35' then multiply .4 by 60 which gives you 24". And 131.41 in that form multiply .41 by 60 which gives you 41' and then .6 by 60 which gives you 36. So the final answer is 48 degrees 35' 24" and 131 degrees 24' 36".

2. 2 tan theta+1=0
Subtract 1 then divide both sides by 2. Theta= tan^-1(-1/2) which gives you 26.565 degrees. Tan is negative so its in the 2nd and 4th quadrant. To move it to the 2nd quadrant make -ve and +180 which gives you 153.435 degrees. To move to the 4th quadrant make -ve and +360 which gives you 333.435. Now convert to degrees minuets seconds and your final answer is 153 degrees 26' 6" and 333 degrees 26' 6".

8-1

5 sec theta +6=0
                  -6 = -6
5 sec theta= -6
/5                 /5
sec theta= -6/5= -33.557= -33degrees 32 mins. 25 sec.
-33.557+180=146.443= 146 degrees 26mins. 34 sec.
33.557+180=213.557= 213 degrees 32 mins 25 sec.

First you have to get sec theta by itself.  So you have to move all the other numbers to to the other side of the equation.  To do this, you have to subtract 6 on both sides.  Then you have 5 sec theta= -6.  Then you divide 5 on both sides and we have sec theta= -6/5.  Now, you have to find the decimal number of sec theta of -6/5.  When you plug that into your calculator, you get -33.557.  Next, you have to figure out what quadrants sec is negative in.  Cos is the recipracal thing of sec, so you know that cos is negative in quadrants two and three.  Now that you figured that out, in the second quadrant, you are supposed to make your number that you figured out negative and add 180 to it.  Doing this you get 146.443.  Then, you figure out what your number is in the third quadrant.  In the third quadrant, you make you number positive and add 180.  You get 213.557.  You cant keep any of your answers you found in decimals, so you have to convert them into degrees, minutes, and seconds.  To do this you just multipy the decimals by 60.

Steps to graph a trig function:

1. Identify if there is a negative or a positive on A after the equation is solved to find which way it will start

2. Find the period , you do this by seeing where it has completed it’s curve and is at the same point on the y axis as where the beginning point of the first curve started. OR use the formula p= 2pi/B

3. Find the amplitude , y=Asin Bx , y= Acos Bx

4. Write the five implied points (coming from your unit circle) : pi/2, pi, 3pi/2, 2pi

5. Divide every one of these points by B of equation

6. If anything is added or subtracted in parenthesis in the standard equation, do the opposite for all five points

7. Sketch your graph by using the latest data (either quotient after dividing or the sum or difference of the parenthesis added or subtracted) in their points then sketch.

Note: if it’s cos, it will not start on the x axis, if it is sin, it will.

stuff.

So, first of all: “α” stands for alpha.

M = tan α : m= slope of a line and α = angle of inclination

tan 2α = B / A-C finds the angle of inclination for a conic.

Ex1 Find the inclination of -3x + 6y = 12.
Step 1: Solve this equation for y.

When you do so, you get y = (1/2) x = 12

Step 2: Use the formula for a line : m =tan α
tan α = (1/2)
arctan(inverse of tan): α= (1/2)
^ plug into calculator

Step 3:
plot where tan is positive and negative (1^st quadrant and 3^rd quadrant)
So α = 26.565 , to get the third quadrant’s value, add 180, so quadrant 3 = 206.565
convert to minutes degrees and seconds:
26 degrees 33 minutes and 54 seconds , 206 degrees 33 minutes and 54 seconds.

Tah dah, and it is complete.

Simple Trig Equations

Chapter 8-1 is pretty much the same as Chapter 7-6, with a few more added notes.

m=tan(theta), where m is the slope of a line.
theta- "alpha" also known as "angle of inclination."

tan2(theta)= B/A-C finds the "angle of inclination" of a conic.
Ax^2+Bxy+Cy^2+...=___

Example 1:
Solve:
3cos(theta)=1
*divide both sides by 3
theta=cos^-1=1/3
*which gives you 70.529
*cosine is positive in the first and fourth quadrants
*make 70.529 negative and add 360 to move it to the fourth quadrant
-70.529+360=289.471degrees
*you must change to degrees, minutes, and seconds
=70degrees31'44'' and 289degrees28'15''

Example 2:
Solve:
6csc(theta)-9=0
*First, add nine to 0.
*divide by 6 on both sides
theta=csc^-1(3/2)
*which would be sin^-1(2/3)
=41.810
*sin is positive in the first and second quadrants
*make 41.810 negative and add 180 to move it to the second quadrant
-41.810+180=138.19degrees
*you have to change both to degrees, minutes, and seconds
=41degrees48'36'' and 138degrees11'24''

Example 3:
Solve:
5sec(theta)+6=0
*subtract 6 from 0
*divide both sides by 5
theta=sec^-1-(6/5)
*which would be cos^-1(-5/6)
=33.557degrees
*cosine is negative in the second and third quadrants
*make 33.557 negative and add 180 to move it to the second quadrant and add 180 to 33.557 to move it to the third quadrant
=176.443degrees and 213.557degrees
*youhave to change both to degrees, minutes, and seconds
=176degrees26'34'' and 213degrees33'25''

Finding trig values in equations

To find the vlaue of an angle in an equation you must first set and solve the equation. This takes back to algebra two rules for solving equations. You first get the trig function by its self then solve for it using calculator or trig chart. Then once you solve for it find two answers depending upon thw quadrant its in. Here is the rules for that:
1.-ve+180 (to go to second quadrant)
2.ve+180 (to go to third quadrant)
3.-ve+360 (to go to fourth quadrant)
To find out wich quadrants you need it depends on the function and if it negative or positive.

ex1. 8 sin + 2 = 4
First solve like equation. subtract 2 from 4. 8 sin =2
Then dive by 8. Sin = 2/8
Its not on trig chart so plug into calculator as an inverse to get an angle measure.
You get 14.476.
Sin is positive in first two quadrant. 14.476 is quadrant one. Now you must find two. Use rules above. So -14.476+180 = 165.524. Now take both answers and properly put them in degrees.
(14 degrees28'33", 165 degrees 31' 26")

ex.2 -2 cos + 1=5
-2 cos=4
cos=4/2 *negative is dropped till end
Calculator gives error when plugged in. This means your answer is undefined.

Solving Trig Functions.

One of the things we learned this week was how to solve trig functions.

First you isolate the trig function. Then you take the inverse of the trig function.

(An inverse finds an angle). Next you use your trig chart or calculator to find answer (only use positive value). And lastly you use quadrants to find the right angle positive or negative with trig function.

There will be at least two answers for each inverse.

Reminder: Q 1 ------> Q 2 make negative ass 180 degrees

Q 1 ------> Q 3 add 180 degrees

Q 1 ------> Q 4 make negative add 360 degrees

Example: 4 tan θ – 5 = 0

First you have to solve for theta. So you add 5 to both sides and you get 4 tan θ = 5. Now you need to divide 4 from both sides so you would get tan θ = 5/4.

Now you need to take the inverse of tan θ which would be θ = tan-1(5/4). Next you plug that into your calculator and you get 51.340 which is in the first quadrant. The other quadrant it’s positive in is 3 so you add 360 to 51.340 and you get 231.34.

You can’t leave your answer like this because Brob doesn’t want us too, so you need to change it to degrees, minutes, and seconds.

51.340

.340 times 60 = 20.4

.4 times 60 = 24

231.34

.34 times 60 = 20.4

.4 times 6 = 24

θ= 51º20’24” and 231º20’24”

Danielle Sharp.

Steps to graphing a trig function

1. Identify if there’s a negative and which way it will start.

2. Find the period

3. Find the Amplitude

4. Write your five imp. Points – 0, pi/2, pi, 3pi/2, 2pi

5. Divide every point by B

6. If anything is added or subtracted in parenthesis do the opposite for all five points

7. Sketch

8. Shift points up or down if anything is added or subtracted outside of parenthesis *it goes the same way

Ex. Graph y=2sin2x

1. Sketch what the graph will look like

2. P = 2pi/b P = 2pi/2 P = pi

3. Amp = 2

4-6. 0 0/2 = 0

Pi/2 pi/2/2 = pi/4

Pi pi/2 = pi/2

3pi/2 3pi/2/2 = 3pi/4

2pi pi/2 = pi

7. Sketch the graph using the five points

8. Shift points if necessary(not necessary in this problem)

Solving Trig Equations

This week we learned to solve trig equations.  To do this, you solve the equation as though you have a regular equation.  Then you find the angle that has that value.  Then you determine what quadrants the angles are.  Finally, you convert the two degree values to degrees, minutes, and seconds.

Solve 4sinѲ=3 for 0˚ ≤ Ѳ ≤ 360˚.  Give answers in degrees, minutes, and seconds.

4sinѲ=3

      (4sinѲ)/4=3/4    divide both sides by 4

     sinѲ= .75

     Ѳ=sin^-1 (.75)= 48.5903

     Sin is positive in quadrant I and II.  So the second angle in quadrant II is 180 - 48.5903 =  131.4097

     Finally, change each angle to degrees, minutes and seconds.

1)      Ѳ=48.5903

48.5903 – 48 = .5903 x 60 = 35.418

35.418 – 35 = .418 x 60 = 25.08

Ѳ = 48˚35’25”

 Ѳ=131.4097

131.4097 – 131 = .4097 x 60 = 24.582

24.582 – 24 = .582 x 60 = 34.92

Ѳ = 131˚24’35”

Jenna Roussel

solving trig functions :)

This week in class we learned about how to solve a trig function.

1.) Isolate the trig function.

2.) Take the inverse of the trig function. (Examples sin^-1, cos^-1)

*** An inverse finds an angle.

3.) Use trig chart or calculator to find answer (only use value)

4.) Use quadrants to find the right angle positive or negative with trig function.

Examples:

1.) solve for theta(q)

5 cos q + 9 = 7

-9 -9

5 cos q = -2

/5 /5

cos q = -2/5

q=cos^-1 (-2/5)= 66.422 (you can’t use this one because cos is negative in quadrant two and three not one.)

so to move to quadrant two from one you have to make negative and add 180 which =113.578 ( this is one of your answers)

to find your second answer you have to take 66.422 and add 180 to get your third quadrant answer, which = 246.422 ( this is your other answer)

q=113.578 and 246.422 ( you can’t leave it like this because it has to be in degrees, minutes and seconds)

.578*60 = 34

.68 * 60 = 40

= 113 degrees 34 minutes and 40 seconds

.422* 60 = 25

.32 * 60 = 19

= 246 degrees 25 minutes and 19 seconds

q = 113 degrees 34 minutes and 40 seconds and 246 degrees 25 minutes and 19 seconds

I don't believe Julian.

I don't believe you. It's clearly not orange. However, since your team won at the movie game, I'll let it slide. α maybe. :P OFF TOPIC. YAAAAY.

Anyway, angles of inclination.
For lines, you will use the formula m = tan(α), where m is the slope of the line.
For conics, you will use tan 2α = (B)/(A-C) where Ax^2+Bxy+Cy^2+...

Your angle of inclination will be α.

For those who forget or never knew, a slope of a line in the form of ax+by = c, is (-a/b). You could also solve the equation for y.
The slope of a line in the form of y = mx+b, is m.

You can use a handy dandy trick that works where A=C for conics. The angle of inclination will be π/4, and 5π/4.

Ex.1

Find the inclination of the line 3x+5y = 8.

• It's a line, so you'll be using the formula m = tanα.
• m=slope, and slope = -a/b, so m = -3/5.
• Plugging in, -3/5 = tanα.
• To solve, you have to take the inverse of both sides.
  tan^-1(-3/5) = α
• tan^-1(3/5) = 30.964, tan is negative in quadrants II and IV.
  
• tan^-1(-3/5) = 149.036 and 329.036
  
• α = 149°2'9'' and 329°2'9''

Ex.2
Identify the graph of x^2-xy+2y^2=2 and find the angle of inclination.

• A = 1, B = -1, C = 2
• B^2-4(A)(C) = -7
• It's negative and A does not equal C, so it's an ellipse.
• You will use tan2α = (B)/(A-C)
• tan2α = -1/-1 = 1
  
• Ignore the 2 in front of α for now and take the inverse.
• tan^-1(1) = 45°
• tan is positive in quadrants I and III.
• tan^-1(1) = 45° and 225°
• NOW you divide by 2.
  
• α = 45/2° and 225/2°

The game.

GG guys.

we gonna find GOLDFISH!

• -m=tan α where m is the slope of a line and α (alpha or goldfish if you prefer)is the angle of inclination




• tan 2α = (B)/(A-C) finds the angle of inclination for a conic. Ax^2 + Bxy + Cy^2 + .. =


• A short cut that they don't want you to know for a conic if A = C then α = pi/4 , 5pi/4

Ex1. To the nearest degree find the angle of inclination for -2x + 4y = 8.
First you have to solve for y. so you get y = (1/2)x + 2
Now you use the formula for a line. m =tan α
1/2 = tan α
arctan(1/2) = α
26.565 degrees and 206.565 degrees
So goldfish equals 26 degrees 33 minutes 54 seconds and 206 degrees 33 minutes 54 seconds

Ex2. Identify the graph of the equation and find the angle of inclination for x^2 + 2y^2 - 3x + y = 11

B^2 -4AC

0 -4(1)(2)

-8

Ellipse

tan 2α = (B)/(A - C)

tan 2α = (0)/(1-2)
tan 2α = 0
arctan(o) = 2α

o = α

Ex 3. Find the angle of inclination. 3x^2 + 5xy + 3y^2 - 5x + y = 4

From the shortcut we know that when A = C the angle of inclination is pi/4 , 5pi/4 so the answer is

α = pi/4 , 5pi/4

AND THAT'S HOW YOU FIND GOLDFISH!

αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα

Inverses of Trig Functions

In this lesson, you use both the unit circle and the six other trig functions.

Before you begin, make sure our calculator is in the correct mode so you get the correct answer.

When you work these problems you must make sure that you get at least a minimum of two answers. Some problems may have more than two answers.

In order to move from quadrant to quadrant:

Quadrant one to Quadrant two: make the number negative and add 180 degrees
Quadrant one to Quadrant three: add 180
Quadrant one to Quadrant four: make then number negative and add 360 degrees

Example 1- Sin^-1(50)=0.766
Sin is positive in only the first quadrant and the second quadrant.

In order to move from quadrant one to quadrant two:
-0.766+180=179.234

Now convert your answers into degrees, minutes, and seconds:
0.766*60=45.96
.96*60=57.6

.234*60=114.04
.04*60=2.4

Your answers are:

0 degrees, 45 minutes, 57 seconds
179 degrees, 14 minutes, and 2 seconds

Inverses in trig functions

This past week in Adv Math H we learned about inverses in trig functions.

-This lesson correlates with the unit circle and the 6 trig functions.

-Every inverse has to have a minimum of two answers.

-You must check the mode on your calculator to see if it is correct before entering a radian or a degree.

To move to different quadrants:

To get from quadrant one to quadrant two-make it negative and add 180 degrees.

To get from quadrant one to quadrant three -simply add 180 degrees.

To get from quadrant one to quadrant four- make it negative and add 360 degrees.

Example:

Tan^-1 ( 25 ) = 87.709

-Tan is positive in the first and third quadrant, you would simply add 180 degrees.

-Your answer would then be 267.709. ← (take and turn into degrees, minutes, and seconds.

.709*60=42.54

.54*60=32.4

Your valid answer is 267 degrees, 42 minutes, 32 seconds.

Trig Inverses.

Last week in class we learn about inverses dealing with trig functions.

You must hit mode first and make sure it is in either radians or degrees at the right moment,if it isn't then all of your answers will then be incorrect.

For all inverses you have to have at least two answers.

In order to move quadrants there is a series of steps you must follow:

If your in quadrant one and you want to get to the 2^nd one-make negative and add 180 degrees.

If your in quadrant one and you want to get to the 3^rd -add 180 degrees.

If you in quadrant one and you want to get to the 4^th -make negative and add 360 degrees.

Ex. Sin ^-1 ( .2 ) = 11.536

Since sin is postive in the first and second quadrant, you would make the decimal a negetive number and add 180 to it.

Your answer would then be -2076.48

You cannot leave the answer in this format,so you would then turn the decimal into minutes and seconds.

-.48*60=-28.8

-.8*60=48

Your new answer would be 2067 degrees, 28 minutes, 48 seconds.

It's only stealing if you're in America.

MAKE SURE YOUR CALCULATOR IS IN DEGREES. Okay, now that we have that settled, this blog will be on the inverse of trig functions. For every inverse, you will have at least two answers.

The first thing to do is plug the problem into your calculator, completely ignoring if the number is negative or positive.

Now you have to look at the problem and decide in which quadrants your answer will be in. (Where is sin negative?, etc.)

Then you have to move your answer to the right quadrants:
1^st quadrant : do nothing
2^nd quadrant: make the number negative and add 180 degrees
3^rd quadrant: add 180 degrees
4^th quadrant: make the number negative and add 360 degrees

Ex. sin^-1 ( -3/4 )

1. Plug into calculator and round to three decimal places: 48.590
2. Sin is negative in the third and fourth quadrant
3. Third quadrant: 48.590 + 180 = 228.590
   Fourth quadrant: -48.590 + 360 = 311.410
4. Convert final answers to degrees, minutes, and seconds:
   228.590 = 228° 35' 24''
   311.410 = 311° 24' 36''

STUFFFFFFFFSS ;)

One of the more common notations for inverse trig functions can be very confusing. First, regardless of how you are used to dealing with exponentiation we tend to denote an inverse trig function with an “exponent” of “-1”. In other words, the inverse cosine is denoted as . It is important here to note that in this case the “-1” is NOT an exponent and so,

.

In inverse trig functions the “-1” looks like an exponent but it isn’t, it is simply a notation that we use to denote the fact that we’re dealing with an inverse trig function. It is a notation that we use in this case to denote inverse trig functions. If I had really wanted exponentiation to denote 1 over cosine I would use the following.

There’s another notation for inverse trig functions that avoids this ambiguity. It is the following.

So, be careful with the notation for inverse trig functions!

There are, of course, similar inverse functions for the remaining three trig functions, but these are the main three that you’ll see in a calculus class so I’m going to concentrate on them.

To evaluate inverse trig functions remember that the following statements are equivalent.

In other words, when we evaluate an inverse trig function we are asking what angle, , did we plug into the trig function (regular, not inverse!) to get x.

So, let’s do some problems to see how these work. Evaluate each of the following.

1.

In other words, we asked what angles, t, do we need to plug into cosine to get ? This is essentially what we are asking here when we are asked to compute the inverse trig function.

There is one very large difference however. In Problem 1 we were solving an equation which yielded an infinite number of solutions. These were,

In the case of inverse trig functions we are after a single value. We don’t want to have to guess at which one of the infinite possible answers we want. So, to make sure we get a single value out of the inverse trig cosine function we use the following restrictions on inverse cosine.

The restriction on the guarantees that we will only get a single value angle and since we can’t get values of x out of cosine that are larger than 1 or smaller than -1 we also can’t plug these values into an inverse trig function.

So, using these restrictions on the solution to Problem 1 we can see that the answer in this case is