8-4

today i am going to teach you about relationships amoung the functions. they have reciprocal relationships, relationships with negatives, pythagorean relationships, and cofunction realtionships. the example today will deal with pythagorean relationships. i will list them below. there are 4 steps that will need to be followed LOOSELY meaning do not follow them completely but follow them as needed.

Step1: Algebra
Step2: Identities- try pythagorean theorem, then move everything to sin and cosnif it will help
Step3: Algebra
Step4: continue with 1-3


EX: (sec B - tan B) (sec B + tan B)

first do algebra and do FOIL:
sec B^2 - sec B(tan B) + sec B(tan B) - tan B^2

then you simpilfy into:
sec B^2 - tan B^2

next is identities: pythagorean
1 + tan^2 x = sec^2 x is the identity so put that in use for:
sec B^2 - tan B^2
and your answer = 1

that is how you show relationships between functions. they are way harder in class so don't think this how easy it is.

HELLO MRS. ROBINSON'S CLASS!!!!!!! Welcome to another of episode of the, yes you guessed it, "The Cory Teaches You How to do Something Show" starring the one and only, me, Cory Costanza. Ask yourself one question. Am i ready to learn? Well golly gosh you better be because i am so ready to teach. Today's episode we will learn about Finding Angles of Inclination. JESUS FRICKEN CHRISTMAS IM SO EXCITED!!!!!! Lets get started shall we.

Find the angle of inclination of:
3x + 6y= 18

1. The formula we will use for this problem is m=tan(alpha)


2. Alpha= angle of inclination


3. In this problem, m= slope. m= -A/B so m= -3/6= -1/2


4. Now we know that tan(alpha)= -1/2


5. YOu need to solve for alpha and than you should get alpha= tan-(-1/2)


6. tan-(-1/2)=26.565


7. Now you need to find which quadrants tan is negative in. It is negative in quadrants 2 and 4.


8. You take 26.565 and make it negative than add 180 to move to Q2 and you get 153.435


9. Than you make 26.565 negative and add 360 to move to Q4 and you get 333.435.


10. 153.435 degrees abd 333.435 degrees are your two answers.

I hope you enjoyed this weeks edition of the "Cory Teaches You How to do Something Show". I would like to give a special thanks to Peter Katerjian for calling me and asking me to send him a picture of my math notes so he could do his blog. Without him calling me and asking me to do that, I never would have remembered to do this magnificent, awesome, superb, incredible, unbelieveable, extraordinary(and a bunch of other adjectives you can think of) blog. Well there you have it folks, thats all for tonights show. Join me next weekend so i can teach you how to do something else. Gee Willickers i cant wait. Goodnight everybody.

Yay,blogs.

This week I’m going to talk about section 8-1. I’m also going to include the last few notes from 7-6 because those are important in this section as well. The trig chart is also useful.

To solve for Ɵ, the steps are going to differ from those of an algebra equation.

1. Isolate the trig function.

2. Take the inverse of said trig function.

*An inverse finds an angle by the way. Thought I’d let you know since I got that wrong on not only my multiple choice test, but also free response.

3. Use trig chart (she wasn’t lying about that thing being used for everything) or calculator to find answer. (only use positive value)

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

There are at least two answers for each inverse.

To move quadrants,

Q1-Q2 make negative and add 180°

Q1-Q3 add 180°

Q1-Q4 make negative and add 360°

You will also need to know the following:

m=tanx where m is the slope α “alpha” AKA “angle of inclination” of a line.

Tan2α=b/a-c finds the angle of inclination of a CONIC.

Ax^2+bxy+cy^2+…..=

EXAMPLE:

3cosƟ=1

First, divide by 3.

CosƟ=1/3

Next, take the inverse.

Ɵ=cos^-1(1/3)

Plug this into your calculator and set up this graph:

*note that you are searching for two positive angles.

Remember you switch from Q1-Q4 by making 70.529 negative and adding 360°.

From here you must convert to degrees minutes and seconds. Your final answer should be

Ɵ=70°31’44”

289°28’15”

And that’s how you do that. Also, Microsoft Word hates me therefore the random highlight and change of font that refuses to be changed.

--Sarah

8-2

so today we are going to learn about the graph of y=sinx and y=cosx
y=sinx starts at zero goes up then back to zero then below zero then back up to the x axis
y=cosx starts around 1 and then goes down to the x axis then below it then back up to the x axis then right above it.
- amplitude is how high or low a curve is from the origin "A"
- to find amplitude
y=a sin bx
y=a cos bx
amplitude can't be negative
-period- how long it takes for a curve to repeat itself. 2pi/b
******************Steps to graphing the trig function**********************

1. i dentify if there is a negative to determine which way the graph will go.

2. find the period

3. find the amplitude

4.write the 5 imput points
0
pi/2
pi
3pi/2
2pi

5. divide everything by "B"

6. if anything is added or subtracted in parenthesis do the opposite for all 5 points

7. sketch

***************EXAMPLE*********************

graph y=2sin 2x

1. graph for sin

2. P= 2pi/B B=2 2pi/2=pi

3. Amp=2

5.
4. 0 /2= 0
pi/2 /2= pi/4
pi /2= pi/2
3pi/2 /2= 3pi/4
2pi /2= pi

7. you would graph the points and connect the lines.

BAM!!!!!!!! done

8-2

Today we are going to learn how to solve a trig function. This requires 8 steps. I’m going to start of by telling you the steps. (For some reason, it kept crashing when I tried to insert a picture.)

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

2. Find the Period.

3. Find the Amplitude

4. Write your five important points. (0, π/2, π, 3π/2, and 2π)

5. Divide every point by “B”

6. If anything is added or subtracted in parenthesis do the opposite for all 5 points

7. Sketch

8. Shift the graph up or down if anything is added or subtracted outside of parenthesis.

Now, we’re going to work and example:

Y=2 Sin 3x

1. There is no negative and it will start like this. (axis, up, axis, down, axis)

2. Use the formula 2π/B to find your Period. 2π/3

3. The Amplitude is the absolute value of the number before the function. Therefore, it is 3

4,5. 0x2=0

π/2x2=3π/2

πx2=π/3

3π/2x2=9π/2

2πx2=2π/3

6. Since there is nothing added or subtracted in parenthesis, you sketch the numbers we just found.

7. Sketch your points using the graph you found in step 1.

And that’s how you solve a trig function J

-Carley

8-1

To find an angle of inclination means to use the formula m=tan(alpha). a slope(m) is given and the steps to solve it are shown in the example.. alpha is also known as the angle inclination. Here is an example.
EX: find the inclination of the line

1) the line 2x + 6y = 9
first, find the slope by putting this in slope-intercept form
y= -1/3x + 3/2
second use the formula
m=tan(alpha)
third solve for alpha
-1/3= tan(alpha) take inverse of tan
alpha= inverse of tan (-1/3)
tan(-1)=18.434
tan is negative in the second and fourth quad
161.565 degrees in the second and 341.566 in the fourth
convert these into degrees minutes and second:
inclination= 161 degrees 33' 54" and 341 degrees 33' 58"

8-1 Angle of Inclination

This week I am going to teach you how to find the angle of inclination. The steps are the same as in section 7-6. Remember, the steps are:
1. Isolate the trig function
2. Take the inverse of the trig function, sin^-1(arcsin), cos^-1(arccos), etc.
3. Either use your trig chart or plug into your calculator to find the answer. *only use the positive value*
4. Use the quadrants to find the right angle measures. You determine which quadrants the different angles are in by using the formulas for the different trig functions.
What is different in this section is that you actually use steps one and two. The formula for angle of inclination of a line is m=tan(alpha), m is the slope. The formula for angle of inclination of a conic is 2d=B/A-C.
Example: 3 cos (theta)=1
1. Cos (theta)=1/3
2. Theta=cos^-1 (1/3)
3. This gives you 70.529 degrees
4. -70.529 degrees+ 360 degrees=289.471 degrees
Then you convert that to degrees minutes and seconds, which gives you:
Theta=70 degrees 31’ 44” and 289 degrees 28’ 15”
That is your final answer.
-Braxton-

8-1

In this section, we learn how to find the angle of inclination. M=tan(a); m stands for the slope of a line and a stands for alpha which is also known as the angle of inclination. To find the angle of inclination of a conic, you use tan2a=B/A-C.

Ex. 1 5cos(theta)+10=7

1. First, you subtract 10 which lives you with 5cos(theta)=-3.
2. Now, you divide 5 by on both sides which leaves you with cos(theta)=-3/5.
3. Now you you solve for theta and take the inverse of cos. Theta=cos-1(-3/5).
4. Now you have to find out what cos-1(-3/5) is and what quadrant it is in. It equals to 53.130 which is in quadrant one. *Remember that you don't put the negative in front of 3/5 when you plug it in your calculator*.
5. You have to find what quadrants cos is negative. Cos is negative in quadrants two and three. Now, in order to move from quadrant one to quadrant two, you have to make 53.130 negative and then add 180. You end up with 126.87 in quadrant two. Now you move from quadrant one to quadrant three. In order to do that, you just add 180 which gives you 233.13.
6. Since the numbers in quadrant two and three are decimals, you have to convert them into degrees, minutes, and seconds. To convert 126.87 to degrees, minutes, and seconds, take .87 and multiply it by 60. That gives you 52.2. Now you take .2 and multiply it by 60 which gives you 12. Now you do the same thing with 233.13. .13x60=7.8 and .8x60=48.
7. Final answer: Theta=126 degrees 52 minutes 12 seconds and 233 degrees 7 minutes 48 seconds.

Ex. 2 Find the angle of inclination of 4x+8y=16.

1. First, you have to know that you are using the formula: M=tan(a).
2. Since the equation is a line, you have to find the slope. M=-A/B=-4/8=-1/2.
3. Now you can plug that into the equation: tan(a)=-1/2.
4. Now you solve for alpha and take the inverse of tan: a=tan-1(-1/2).
5. You find out what tan-1(-1/2) equals to: 26.565. Now you find out what quadrants tan is negative in: two and four.
6. Find quadrant two: -26.565+180=153.435. Find quadrant four: -26.565+360=333.435.
7. Turn both of those answers into degrees, minutes, and seconds: .435x60=26.1 .1x60=6 (I only did it once since both of the decimals end with .435).
8. Final answer: A = 153 degrees 26 minutes 6 seconds and 333 degrees 26 minutes and 6 seconds.

Ex. 3 Identify the graph of the equation x^2-5xy+8y^2=1 and find the angle of inclination.

1. First, you have to find out what kind of conic this equation is: B^2-4(a)(c)=(-5)^2-4(1)(8)=168. Since it is a positive number, it is a hyperbola.
2. Plug the numbers from the equation above ^^^^ into tan2a=B/A-C: tan2a=-5/1-8=-5/-7=5/7.
   
3. Now you solve for alpha and take the inverse of tan: 2a=tan-1(5/7).
4. Find out what tan equals to and what quadrant tan is positive in: tan-1(5/7)=35.538. Tan is positive in quadrant one and three.
5. Find out what quadrant three is: 35.538+180=215.538.
6. 2a=35.538;215.538. Divide by two on each side: a=17.769;107.769. Put in degrees, minutes, and seconds: .769x60=46.14 .14x60=8.4
7. Final answer: a=17 degrees 46 minutes 8 seconds and 107 degrees 46 minutes 8 seconds.

-Amber

Sine & Cosine Curves

I'm going to find the Amplitude and period of this function and sketch it's graph.
y=4 cos 2x
The Amplitude(A) is 4 because the number in front of either cos/sin is the A
The period is found by: 2times pie divided by B, which  equals pie(3.14...) because the twos cancel.

Now, To sketch the graph one must do the following steps:

1. Determine the general curvature of the equation; for this equation it will go something like this:(This being the graph in blue)

2.  Next find the period, which we already know is pie

3. Find the amplitude which we also already know is 4.

4/5/6. multiply each point by the number in front of x

0 = 0

    pie/2 = pie/4

pie = pie/2

3*pie/2 = 3*pie/4

two*pie = pie

There is no step 6 in this problem, so on to step 7, which is to graph the equation

Once that is done, you should get this as a graph:

8-2 Graphing Sine and Cosine Curves

Graphing Sine and Cosine Curves

Steps to graphing a trig function:

Before you begin your steps in graphing a trig function, you must first identify the graph as sine or cosine. Sine will start at the origin, going up or down, and cosine will start above, going down, or below it, going up.

1. You must first identify if there's a negative, and which way it will start.
   
   



2. Find the period: 2pi/B





3. Find the amplitude: y=A sinBx y=A cosBx *Amplitude cannot be a negative. *A negative infront of sin or cos flips the curve.





4. Write your 5 imp. points: 0, pi/2, pi, 3pi/2, 2pi





5. Divide every point by "B".





6. If there's any number that's added or subtracted in parenthesis, do the opposite for all 5 points.





7. Sketch your graph.





8. Shift points up or down if anything is added or subracted in parenthesis. *The shift goes in the same direction.

Example : Graph y=3 sin2x

1. the graph is sin starting at the origin, going up





2. period= 2pi/B: 2pi/2= pi





3. amplitude= 3





4. /5./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 2pi/2= pi

7.)

--Jordan Duhon:)

8-2 Graphing Trig Functions!

So, this week I am going to teach you how to graph a trig function.

Graphing trig functions have 8 steps but you don't always need to use all of them. These 8 steps are:

1. First you are going to identify if there is a negative and you are going to figure out which way the graph will start.
2. The next thing you need to do is find the period. A period is how long it takes for a curve to repeat itself. To find a period you use the formula 2Pi/B.
3. The next step is to find the amplitude. An amplitude is how high or low a wire is from the origin. To find an amplitude you just look for the number in front of sin, cos, etc.
4. Once you do all of those steps are going to list your 5 important points, which are always going to be 0, Pi/2, Pi, 3Pi/2, and 2Pi.
5. The next step is to divide your important points by B.
6. The next step you my not always use. The only time you would use this is if anything was added or subtracted in parenthesis in your problem. If that happens then you would do the opposite for all 5 of your important points.
7. Now you would sketch the graph.
8. We never actually got to this step yet and we never used it but I'm going to tell you it anyway. You would use this step is anything was added or subtracted outside of the parenthesis. If anything was then you would shift points up or down.

NOTE: Amplitude cannot be negative. A negative in front of sin or cos flips the curve.

Now I am going to do an example of these types of problems.

Example: 4 sin x

1. There are no negatives in this problems and the graph should look like this:
2. P=2Pi/1 P=2Pi
3. Amplitude=4
4. 0, Pi/2, Pi, 3Pi/2, 2Pi
5. You would divide them all by B which is 1. So they will all stay the same.
6. There are no numbers in parenthesis being subtracted or added.
7. The sketch of the graph should look like this:

8. You don't have anything to do for this step.

And that is how you graph trig functions!

--Halie! :)

8-1 finding angle of inclination

today i am going to teach you how to find and angle of inclination. first you must know the formula before you can do it. the formula is: m=tan(alpha). you will be given a slope and then the steps will be given and followed in order to find the angle of inclination. alpha is also know as the angle inclination which would be good to know when follwing the steps to finding it. here is an example from one of the homework problems this week.

EX: find the inclination of the line

1) the line 3x + 5y = 8
first you find the slope by making this slope intercept form
y= -3/5x + 8/5
second is you right out your formula
m=tan(alpha)
third is solve for alpha
-3/5= tan(alpha)----take inverse of tan
alpha= inverse of tan (-3/5)
this =30.964
tan is negative in the 2nd and 4th quadrant
you get 149.036 and 329.036
convert to degrees minutes and second and u get:
inclination= 149 degrees 2' 6" and 329 degrees 2' 6"


that is all for today

7-6 Inverse Trig Functions

This week I am going to teach you how to solve inverse trig inverse trig functions. When you find the inverse of a trig function you are finding an angle measure. In just four simple steps, you can solve these problems.
Steps:
1. First of all, you have to isolate the trig function
2. Then you have to take the inverse of the trig function, sin^-1(arcsin), cos^-1(arccos), etc.
3. You can do one of two things in this step. You can either use your trig chart or plug into your calculator to find the answer. *only use the positive value*
4. Lastly, you use the quadrants to find the right angle measures. You determine which quadrants the different angles are in by using the formulas for the different trig functions.
*each problem will have at least two answers*
Moving Quadrants:
• Quadrant I to quadrant II-----make the number negative and add 180 degrees
• Quadrant I to quadrant III----add 180 degrees to the number
• Quadrant I to quadrant IV----make the number negative and add 360 degrees
Example: sin^-1 (33/100)
1. This step is already done
2. This step is also already completed
3. You have to plug it into your calculator because 33/100 is not on the trig chart. This gives you 19.269 degrees. *sin is positive in quadrant I and quadrant II, so that is where your two answers will be located*
4. 19.269 degrees is your first answer because it is in quadrant one. Then you make it negative and add 180 degrees to move to quadrant II. This gives you 160.731 degrees (remember, your answer cannot be a decimal, so you have to convert the numbers into minutes and seconds)
Final Answer: 19 degrees 17 minutes 45 seconds, & 160 degrees 43 minutes 51 seconds

-Braxton-

This week I’ve decided to talk about section 7-5, the other trig functions. This is probably the easiest thing we did in all of chapter seven. The processes in this section use the unit circle, the trig chart, and your knowledge of trig functions. So, if you don’t know those things you should look them over before trying this section (:

-First, you should know how to use your calculator. You have special buttons for sin, cos, and tan, but not for csc, sec, and cot. You need to know the following:

cotƟ=1/tan( )

secƟ=1/cos ( )

cscƟ=1/sin( )

EXAMPLE :Find the other 5 trig functions given tanƟ= -24/7 ∏<Ɵ<∏/2

First, you have to determine which quadrant your graph will fall in. It is easiest to do this by drawing a graph like this:

Next, you refer to your unit circle. ∏ is at (-1,0) and ∏/2 is at (0,1). Therefore, your graph will be in the second quadrant. You would then draw in your triangle and your graph will look like this:

Sincs tan=y/x, you know that the leg of the triangle on your y axis will be 24 and the leg of your triangle on the x axis will be seven. You can then find the hypotenuse using the Pythagorean Theorem. (a^2+b^2=c^2). NOTE: You do not have to use the Pythagorean Theorem if you know your triplets. Your triplets are 3,4,5 and 5,12,13 and of course 7,24,25. After determining that your hypotenuse is 25, your graph will look like this:

Based on this information, your answer will be as follows (keep in mind that your x is negative and your hypotenuse is your radius):

I like to put this here to remind myself: x= -7, y=24, r=25

sinƟ=y/r= 24/25

cscƟ=r/y=25/24

cosƟ=x/r= -7/25

secƟ= r/x= -25/7

cotƟ=x/y= -7/24

Remember that tanƟ=y/x= -24/7 was the given function J

--Sarah

Finding cot, csc, and sec

Finding cotangent, cosecant, and secant is not so difficult. All one has to do to solve for an inverse trig function is enter in their calculator in fraction form 1/(whatever the inverse of the regular trig function is)

1/tan= cotangent

1/sin=cosecant

1/cos=secant

Examples:

1. Find the cot(1.25)

2. Plug in calc as 1/tan(1.25)

3. The answer should be, rounded to 3 decimal places, 45.829 degrees

1. Find the csc(3.45)

2. Plug in calc as 1/sin(3.45)

3. The answer is 16.618 degrees

1. Find the sec(2.32)

2. Plug in calc as 1/cos(2.32)

3. The answer is 1.00 degree.

Finding Sec, Csc, and Cot

All right, so I don't have my math book or notes for this Blog, so this is the best I could come up with off the top of my head :D!

First I am going to find Sec(73degrees)
1. Enter 1 divided by Cos(73 degrees) into your calculator
2. Hopefully get 3.42 as an answer
3. 3.42 is your final answer

Second, I will find Csc(97degrees)
1. Enter 1 divided by Sin(97degrees) into your calculator
3. Get 1.008 as an answer
3. 1.008 is your final answer

Third, I will find Cot(28degrees)
1. Enter 1 divided by tan(28degrees) into your calculator
2. Get 1.881 as an answer
3. 1.881 is your final answer

(All of the angles used in these examples were random and this process will work for any given angle)

Hello everybody, and welcome back to another episode of the "Cory Teaches You How to do Something Show"!!! Please hold your applause until the end of the show. On this weeks show, I(Cory Costanza) am going to teach you how to do something. Yes yes i know, it is so fulfilling when you learn something and believe me im just as fulfilled to teach you. Now, first i must think of what i can do this blog on because i dont remember doing anything this week except taking tests and quizzes. Now i know your thinking, "Gee Cory, dont you have your advanced math binder with all your stuff from the week in it?". To answer that question, NOO I DO NOT!!!! I left it in my locker so im going to have to wing it on this weeks episode. HMMMMMMM, how about we learn about "FINDING ALL SIX TRIG FUNCTIONS" since i remember that being on the test this week.

I will not write the steps out this week, i will explain each step instead as I am going.

Ex: Given point (4,3. Find all six trig functions

1. First you would draw your triangle in the first quadrant.


2. Next, you need to find the hypotenuse value of the triangle. Do this by using the pythagorean theorem. 4^2 + 3^2=c^2


3. After you solve for c, you should get c=5


4. Now you know that x=4 y=3 and r= 5


5. Now it is just a matter of plugging these numbers into the trig functions


6. sin= y/r=3/5


7. cos=x/r=4/5


8. tan=y/x=3/4


9. csc=r/y=5/3


10. sec=r/x=5/4


11. cot=x/y=4/3

Well there you have it folks, I taught you how to do something else. Boy ol Boy i cant wait till next weekend to teach you something else. I may have to start charging for all these free tutoring lessons. Anywho, thats all for this weeks edition of the "Cory Teaches You How to do Something Show".

7-5 Other Trig Functions

7-5 Other Trig Functions

In order to find the three trig functions that aren't on your calculator already, you simply plug in the following formulas: cot x= 1/tan( ) sec x= 1/cos( ) csc x= 1/sin( )

-Ex: Find the value to 4 significant digits.

a. cot 6= 9.514
You plug in your calculator: 1/tan(6). You should get 9.514364454. Now you round to 4 significant digits, and get 9.514

b. csc 54= 1.236
Plug in: 1/tan(54). You get 1.236067977. Round, and get 1.236

c. sec 15= 1.035
Plug in 1/cos(15). You get 1.03527618. Round, and get 1.035

Now, I will show you how to find the other 5 trig functions when given a certain trig function, and which quadrant it's located in.

Example: Find the other 5 trig functions given tan x= 3/4, Pi
-First I will locate the quadrant. It's in quadrant 4 because it is greater than Pi (-1,0), and less than 2Pi (1,0).

-Now, you determine the radius, because you already have y and x(tan=y/x=3/4).

-Now, you plug in the pythagorean theorem: 3^2+4^2=c^2, 9+16=c^2, 25=c^2. You should get 5(square root of 25).

-You can now find the other 5 trig functions. (y=3, x=4, r=5)

sin= 3/5 cos= 4/5 tan= 3/4
csc= 5/3 sec= 5/4 cot= 4/3

---Jordan Duhon:)

7-6 Inverse Trig Functions!!!!!

Sooo today im going to be doing my blog on inverse trig functions since that is the only thing new we learned this week.

there is four different steps to solving inverse trig functions in the section.

-the first step in solving inverse trig functions is to isolate the trig function.

-The second step is to take the inverse of the trig function-> (sin^-1)=arc sin (cos^-1)=arc cos

************an inverse finds an angle*********** ALWAYS!!!!!! ha

-The third step is to use either the trig chart or plug into your calculator to find the answer.

(your only going to use the positive value)!!!!

-The last step you use quadrants to find the right angle measure. you will determine if it is positive or negative by usuing the trig chart.

******there will be atleast TWO answers for each inverse functions********

- to move to quadrants**

q1 ->q2-- make negative and add 180 degrees

q1 ->q3-- add 180 degrees

q1 ->q4-- make negative and 360 degrees

Now that i have deeply explained in details to you how you mathematically solve this inverse trig function we will now proceed to a physical mathematical inverse equation :)

This will be an example.

cos^-1 (3/4)

1. as you can tell step one has already been completed.

2. this step is also already done.

3. 3/4 is not on the trig chart, so when you plug the problem into your calculator you get 41.410 degrees.

4. 41.410 degrees is quadrant 1 so that is going to be your first answer. When you turn that into degrees minutes and seconds you get 41 degrees 24 minutes and 36 seconds. Next you would move 41.410 degrees to the 4th quadrant, so you would turn the degree negative and add 360, so you would get 318.59, which would be 318 degrees 35 minutes and 24 seconds.

***your answers would then be, 41 degrees 24 minutes 36 seconds and 318 degrees 35 minutes 24 seconds.

and that is how you do inverse trig functions.

thanks for reading.

-Brad!

7-5 The other trig functions

Section 7-5 is the same as section 7-4, but with tan, cot, sec, and csc, instead of using sin and cos. It is not that hard since you are pretty much doing the same thing as 7-4.

First off you need to know what does tan, cot, sec, and csc equal too. Tan equals to y/x. Cot equals to x/y. *Notice how Tan and Cot are opposite from each other. Sec equals to r/x which is the opposite of cos. Csc equals to r/y and is the opposite of sin.

Since this section is pretty much the same as section 7-4, I won't give the steps. I will just explain as I go. I will do an example for each function and do two negative degrees and two positive degrees.

Ex. 1. Tan 210 degrees

1. 210 is in quadrant three.
2. Tan = y/x = +ve/-ve = -ve. It is negative because in the third quadrant the y coordinate is positive and the x coordinate is negative. If you divide a positive by a negative, it will give you a negative.
3. Now to find the reference angle, you will subtract 180 from 210. So 210 - 180 = 30.
   
4. So the reference angle for Tan 210 degrees is -Tan 30 degrees.

Ex. 2 Sec 125 degrees.

1. 125 is in quadrant two because it is between 90 and 180 degrees.
2. Sec = r/x = 1/+ve = +ve. It is positive because the x coordinate is positive and when you divide 1 by a positive you get a positive number.
3. Now you have to find the reference angle. 125-180 = -55. Since it is negative, you have to take the absolute value of it. The absolute value of -55 is 55.
4. The reference angle for Sec 125 degrees is +Sec 55 degrees.

Ex. 3 Cot -220 degrees.

1. -220 is in quadrant two. Since it is negative you go backwards. So 90 degrees would be quadrant four, 180 degrees would be quadrant three, 270 degrees would be quadrant two, and 360 degrees would be in quadrant one.
2. Cot = x/y = +ve/-ve = -ve.
3. You do not want to use the -220 to subtract from 180 because that would just give you a bigger number. So you use 220 and subtract from 180 which will give you 40.
4. The reference angle for Cot -220 degrees is -Cot 40 degrees.

Ex. 4 Csc -150 degrees

1. -150 is in quadrant three. Like I said in the example before this one, you go backwards since it is negative.
2. Csc = r/y = 1/-ve = -ve.
3. Again, like I said in the example from above don't use the -150 to subtract. So 150-180 = -30. Since it is a negative answer, you take the absolute value of it which gives you 30.
4. The reference angle for Csc -150 degrees = -Csc 30 degrees.

That is how you find a reference angle for the other trig functions, it is not that hard since in the last section, we learned how to find a reference angle for sin and cos. It is the same thing, just with different trig functions.

-Amber :)

7-6 Inverse Trig Functions

Okay, so today I am going to explaing to you how to solve inverse trig functions. There are four easy steps to solve these problems.

These four steps are:

1. The first thing you would do is isolate the trig function.
2. The next thing is to take the inverse of the trig function, for example sin would be sin -1.
3. You would then use the trig chart or calculator to find the answers.
4. The last step is to use the quadrans to find the right angle.

Note: There is always going to be at least two answers for each inverse.

--When you move quadrants to find your answers you would do the following:

• To go to quadrant 1 to 2, you would make the angle negative and add 180 degrees.
• To go to quadrant 1 to 3, you would just simply add 180 degrees.
• To go to quadrant 1 to 4, you would make the angle negative and add 360 degrees.

Now I am going to do an example of these types of problems.

Example 1: sin-1(.9)

1. The trig function is already isolated.
2. It already gives you the inverse of the function, so you don't need to do anything for this step.
3. Since .9 is not on the trig chart you would plug that into your calculator to find the degree. You would then get 64.158 degrees.
4. Since sin is y/r and .9 is positive, you are going to find angles in quadrant 1 and 2. 64.158 degrees is already in quadrant one so all you would have to do is convert that into degrees, minutes, and seconds, which would be 64 degrees 9 minutes and 28 seconds. Next you would want to find an angle in quadrant 2. According to the note above, to move from quadrant 1 to 2 you would make your angle negative and add 180. This would then give you 115.842, you would then convert that into degrees, minutes, and seconds, which would give you 115 degrees 50 minutes and 31 seconds.

Your answer would then be your two angles which are: 64 degrees 9 minutes 28 seconds and 115 degrees 50 minutes 31 seconds.

--Halie!

7-6

I'm going to teach you to solve for ø. The steps are different than an algebra equation.

1. Isolate the trig function.

2. Take the inverse of the trig function (arc sin)sin–1 (arc cos)cos -1

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

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

There are at least two answers for each inverse.

To move quadrants:

• Q1àQ2 make negative add 180º
• Q1àQ3 Add 180º
• Q1àQ4 Make negative add 360º

Now I'm going to show you an example using the steps above.

Cos^-1 (.2)

1. The trig function is already isolated.

2. The inverse is already taken.

3. Since .2 is not on the trig chart, you have to change it into degrees. You would then get 11.46. This means that it is in the quadrant I.

4. Because Cos is x/r, it would be in quadrants I and III. Since we already know the degree in Quadrant I is 11.46 we have to find the degree in quadrant III, to do that you would at 180º. so 11.46º+180º=196.46º. Now we have to change the degree into degrees, minutes and seconds.We learned this in the first section. So for 11.46, you would get 11º27'36" and 196.46º would be 196º27'36".

Your final answer would be Cos^-1 (.2)=11º27'36" and 196º27'36"

-----Carley Lynn Guidry-----

7-5 other trig functions

today im going to show you how to find refernce angles with other trig functions. in 7-4 we learned how to do this with sin and cos which was fairly easy. today is differnt. today will be showed how to do it with tan cot sec and csc. steps 1 and 3 are the same. step 2 however is slightly differnt. insted of y/r for sin and x/r for cos, now we have to do y/x for tan and x/y for cot to find out if its positive or negative. we also have r/y for csc and r/x for sec. it is not much differnt but it could get tricky.

EX:
a) tan 123 degrees
1) it is b/w 90 and 180 degrees so it is the second quadrant
2) it is negative because it is y/x. y is positive and x is negative making it a negative tan.
3) 123-180=-57 |-57|= 57
refernce angle is -tan 57


b) cot 310 degrees
1) it is b/w 270 and 360 degrees making it in the 4th quadrant.
2) it is negative because it is x/y. y is negative and x is negative making it a negative cot.
3) 310-360=-50  |-50|=50
refernce angle is -cot 50


c) csc -300 degrees
1) -300 is in 1st quadrant because you go backwards 300 degrees in the unit circle
2) it is positive because it it r/y. r is positive and y is positive making it a positive csc.
3)-300+360= 60
reference angle is csc 60

d) sec 245
1) it is b/w 180 and 270 making it in the 3rd quadrant.
2) it is negative because  it is r/x. r is positive and x is negative making it a negative sec.
3) 245-180= 65
reference angle is -sec 65

Finding Reference Angles

Hi, Welcome to another episode of the "Cory Teaches You How to do Something Show" (applause). On this weeks show, I(Cory Costanza) will teach you how to do something(more applause). I hope your ready to learn because boy o boy am i ready to teach. Lets learn about "Finding Reference Angles". Golly Gosh im excited.

These are the steps 0n how to find a reference angle, and i beg you not to fall asleep.
1.) Find the original quadrant and sketch
2.) Determine if the angle is positive or negative using unit circle methods.
3.) Subtract 360 or 180 degrees until the absolute value of theta is b/w 0 and 90 degrees

Lets work an example problem. Well jesus christmas that sounds like such a good idea.
Find the reference angle for:
sin1055

1.) The originl quadrant for this probelm is quadrant 4.
2.) Find out if it is positive or negative by using the formula y/r because it is sin. r is always 1. In quadrant 4, y is negative so no you know the answer is negative.
3.) Now subtract 360 from 1055 until the absolut value is b/w 0 and 90.
1055-360-360-360=-25
The absolute value of -25 is 25
The final answer is -sin25

Well there you have it folks. This is how you find reference angles. I hope you enjoyed reading this as much as i enjoyed typing it up for you. Tune in next week on the"Cory Teaches You How to do Something Show"(applause) to see what I(Cory Costanza) teaches you next. I cant wait to see you again.

Trig Chart

Nothing compares to how important it is to learn this trig chart if one wants to be successful in higher level mathematics.

sin 0= 0

sin pi/6=1/2

sin pi/4= square root of 2/2

sin pi/3= square root of 3/2

sin pi/2= 1 cos pi/2= 0

cos 0= 1

cos pi/6= square root of 3/2

cos pi/4= square root of 2/2

cos pi/3= 1/2

cos pi/2= 0

csc 0= undefined

csc pi/6= 2

csc pi/4= square root of 2

csc pi/3= 2 square root of 3/3

csc pi/2= 1 sec pi/2= undefined

sec 0= 1

sec pi/6= 2 square root of 3/3

sec pi/4= square root of 2

sec pi/3= 2

sec pi/2= undefined

tan 0= 0

tan pi/6= square root of 3/3

tan pi/4= 1

tan pi/3= square root of 3 cot pi/3= square root of 3/3

tan pi/2= undefined cot pi/2= 0

cot 0= undefined

cot pi/6= square root of 3

cot pi/4= 1

cot pi/3= square root of 3/3

cot pi/2= 0

As a reminder, pi/6= 30 degrees, pi/4= 45 degrees, pi/3= 60 degrees, pi/2= 90 degrees

Also, sin=sine, cos=cosine, csc= cosecant, sec=secant, tan=tangent, cot=cotangent

--Sameer

7-4 Reference Angles

This week I am going to teach you how to find reference angles. It only takes three easy steps.
Steps:
1. You have to find the original quadrant (using your unit circle) and sketch.
2. Then you must find out if the angle is positive or negative by using your unit circle. *Use the following formulas to determine positive or negative: sin=y/r, csc=r/y, cos=x/r, sec=r/x, tan=y/x, and cot=x/y*(r is always 1)
3. Lastly, you subtract the angle measure by 360 degrees or 180 degrees until the absolute value of theta is in between 0 degrees and 90 degrees.
Examples:
1. sin 128 degrees 1. It is in quadrant II because 128 degrees is in between 90 degrees and 180 degrees.
2. The formula for sin is y/r, so you get (y/r)= -/1. This gives you a negative answer.
3. Then you do 128 degrees- 180 degrees. This gives you -51 degrees.
The answer is –cos 51 degrees. Remember when you put the answer to step number three in your final answer, you take the absolute value of that number.
Now you know how to find reference angles.
-Braxton-

7-4 Reference Angles

Today we are going to learn how to find a reference angle. This is done by using 3 simple steps.

Steps:

1. Find the original quadrant and sketch.

2. Determine if the angle is positive or negative using unit circle methods.

3. Subtract 360º or 180º until the absolute value of ø is between 0 and 90º

Now I am going to show you an example of how this is done.

Find a reference angle for Sin(355º)

1. Sin(355º) is in quadrant 4 because it is between 270º and 360º.

2. Because we are trying to find a reference angle for sin, you use the formula y/r to determine whether or not it's going to be negative or positive.

335º is in Quadrant 4, so you use the formula y/r to find out whether or not it's negative.Y is negative because you're in quadrant 4 and r is always 1, so negative/1=negative.

3. 355º-360º=|-5º|

This means that Sin(355º)= -Sin(5º)

Your final answer will be written as Sin 355º= -Sin 5º

----Carley(:

7-4 Reference Angles

Reference angles have to be between 0 and 90 degrees.

Steps:

1. Find the quadrant then sketch.
2. Find out if the angle is + or -.
3. Subtract either 360 degrees or 180 degrees until the absolute value of theta is between 0 and 90 degrees.
   

Ex. 1 Find the reference angle for sin400 degrees.

1. First, you have to subtract 180 from 400 to find out what quadrant it is in. When you subtract you get 220. That is in the third quadrant.

2. To find out if is it + or -, you do y/r. The radius is always one. So you end up with -/1 which is -. The reason the top number is - is because the y coordinate in the third quadrant is -. So it is going to be -.

3. Now to find the reference angle of this problem, you have to subtract 220 from 180. When you subtract you get 40.

Your final answer is sin400 degrees = -sin40 degrees.

Ex. 2 Explain cos7pi/4 as a reference angle.

1. First of all you need to put the problem in degrees. To do that you take 7pi/4 and multiply it by 180/pi. When you do this pi will cancel and leave you with 315 degrees. So now your problem is cos315 degrees. Now we can find what quadrant it is in. It is in quadrant four.

2. Now you need to find out if it is + or -. Cos is x/r. So x/r = +/1 = +.

3. Now you have to subtract 315 from 360 which gives you -45. You then take the absolute value of that which is 45.

Your final answer is cos315 degrees = +cos45 or cos7pi/4 = +cospi/4.

Ex. 4 Evaluate sin60 degrees.

The only thing you have to do for this problem is go to your trig chart and find sin60 degrees. According to the trig chart, sin60 degrees is the square root of 3/2. So sin60 degrees = square root of 3/2.


-Amber :)

7-4 Reference Angles

Finding Reference Angles Using Sin and Cosine

In order to find a reference angle, it must be between 0 degrees and 90 degrees.
There are 3 steps used in finding a reference angle:

1) Locate the original quadrant then sketch.

2) Decide whether the angle is positive or negative using the Unit Circle.

3) Subtract either 360 degrees or 180 degrees until your absolute value is between 0 degrees or 90 degrees.

Example 1: Find a reference angle for sin 1000 degrees.

*Before I can find the quadrant, I must subtract: 1000-360(2)=280. Now I can state the quadrant.

1) It is in quadrant four because 280 is between 275 and 360.

2) It is negative because sin is negative if it's in the third or fourth quadrant.

3) 280-360=-80

-The absolute value of -80 is 80.

answer: -sin 80 degrees

Example 2: Find a reference angle for cos (-50 degrees).

1) It is in quadrant four because -53 falls between -90 and 0.

2) It is positive because cos is positve if it's in the first and fourth quadrant.

3) Since -53 is already between 0 and 90, all you do is find the absolute value.

answer: cos 53 degrees

----Jordan Duhon

The Trig Chart :)

Soo Brob its Danielle on Sarahs once again..

Sooooooooooooooooooooooo this week we learned the trig chart. IT IS VERY IMPORTANT and we will need it for the rest of our mathematical lives. The trig chart teaches us trig functions.

Sin over cos gives you tan

Key point: reciprocal of sin is csc

Reciprocal of cos is sec

Reciprocal of cot is tan (cot= sin over cos)

Sin0= 0

Sin 30 degrees pi over 6= ½

Sin 45 degrees pi over 4= square root of 2 over 2

Sin 60 degrees pi over 3= square root of 3 over 2

Sin 90 degrees pi over 2= 1

cos0= 1

cos 30 degrees pi over 6= square root of 3 over 2

cos 45 degrees pi over 4= square root of 2 over 2

cos 60 degrees pi over 3= 1/2

cos 90 degrees pi over 2= 0

csc0= undefined

csc 30 degrees pi over 6= 2

csc 45 degrees pi over 4= square of 2

csc 60 degrees pi over 3= 2 square of 2 over 3

csc 90 degrees pi over 2= 1

sec0= 1

sec 30 degrees pi over 6= 2 square root of 3 over 3

sec 45 degrees pi over 4= square root of 2

sec 60 degrees pi over 3= 2

sec 90 degrees pi over 2= undefined

tan0= 0

tan 30 degrees pi over 6= square root of 3 over 3

tan 45 degrees pi over 4= 1

tan 60 degrees pi over 3= square root of 3

tan 90 degrees pi over 2= undefined

cot0= undefined

cot 30 degrees pi over 6= square root of 3

cot 45 degrees pi over 4= 1

co 60 degrees pi over 3= square root of 3 over 3

cot 90 degrees pi over 2= 0

Example 1: cos 45°

So you go look on your trig chart and the answer is square root of 2 over 2

Example 2: sin 210°

First you have to subtract by 360° because 210° isn’t on your trig chart, and you get 30° so you look on your trig chart and it the answer would be ½

Example 3: cos (-30)

You look on your trig chart and you get square root of 3 over 2

P.S Sarah hates the trig chart.

---Danielle