Week 1 Blog Prompt

Explain the relationship between the following trig functions:

a. Sine and Cosine
b. Tangent and Cotangent
c. Sine and Cosecant
d. Cosine and Secant
e. Sine, Cosine & Tangent

Section 7-1

This week in Advanced Math we started in Chapter 1. In section 1, we learned about the measurement of angles. To find the measure of an angle in either degrees or radians and to find coterminal angles. Angles can also be measured in decimal degrees. To convert between decimal degrees and degrees, minutes, and seconds. For example,

12.3 = 12 +0.3(60)' = 12 18'

25, 20',6''=25+(20/60) + (6/3000)= 25. 335

I learned within the past two weeks in advanced math, if you know all of your formulas then everything else comes easy.

7-4 Finding a Reference Angle

Last week in class, the advance math students learned how to find reference angles. At first, the concept seemed difficult to grasp, but after days of practice the concept almost seemed similar to second nature.Here is a step by step list on how to solve

- To Find a Reference Angle

1.Find the Quadrant the angle is in


2.Determine if the trig function is positive or negative


3. Subtract 180 degrees from the angle until the absolute value is between 0 and 90 degrees.


4. If it is a trig chart angle ,plug in. If not, leave it or plug it in a calculator.


*Hint: Evaluate means number answer unless otherwise is specified.



Trig Chart: 0-0 degrees, 30 degrees- pi/6,45 degrees- pi/4, 60 degrees-pi/3, 90 degrees-pi/2




Ex: Find the Reference Angle for SIN 695 degrees.

1st step: Locate the quadrant. Since its 695 degrees, it'll be located in the 4th quadrant which is - +. 2nd step: you have to subtract 360 from 695 that will get you 335(which is the coterminal)

3rd step: you subtract 180 from 335 until u get -25.

Since 695 is in the 4th quadrant..sin 695<0 695 =" -sin25">

This week in advanced math we started in chapter seven. The section I understood the most was 7-1 Measurement of angles. We learned how to convert radians to degrees, and degrees to radians. We also learned how to find coterminal angles. I understood more the first week of school than the second week of school.

If given a number in radians to be converted into degrees, this is the formula: # x 180/pi

If given a number in degrees to be converted into degrees, this is the formula: # x pi/180

Example # 1: Convert 55 degrees to radians
1. Plug in to the formula 55 x pi/180
2. Treat pi as a variable.
3. You get 55pi/180
4. Divide 55/180
5. You get 11pi/180

Example # 2: Convert 3pi/4 radians to degrees
1. Plug into the formula 3pi/4 x 180/pi
2. The pi is treated as a variable and they cancel out.
3. You then get 540/4
4. Your answer is 135 degrees.

That is what I understood most in the past two weeks in this class. The key to understanding everything in this class is the formulas.

Section 7-1

Section 7-1

This week in Advanced Math we started on Chapter 7. We started with section 7-1 and we learned the two units of measure are degrees and radians. We converted degrees to radians, radians to degrees, and found coterminal angles. All of these problems can be solved with the right formula. Let’s start with some example problems to understand more about what we learned.

EXAMPLE 1: The formula for converting degrees to radians is degrees multiplied by pi over 180 (degree * (pi/180)).

Convert 60 degrees to radians
1. Plug into formula - 60 * pi/180
2. Multiply to get - 60pi/180
3. Plug into your calculator - 60/180 (You do not need to plug pi into the calculator)
4. Your final answer will be - .33 as a fraction it is 1/3

EXAMPLE 2: The formula for converting radians to degrees is radians multiplied by 180 over pi (radian * (180/pi))

Convert pi/5 radians to degrees
1. Plug into formula - pi/5 x 180/pi
2. Multiply and the pi’s cancel each other out - 180/5
3. Plug into your calculator - 180/5
4. Your final answer will be - 36 degrees

Finding negative and positive coterminal angles is the simplest part of this whole section. To find a degree's positive coterminal angle, add 360. To find a negative coterminal angle you subtract 360. However, if your number is in radians you either add 2pi or subtract 2pi to get a positive or negative number. Let’s try some examples.

EXAMPLE 3: Adding and Subtracting coterminal angles with 360

Find a positive coterminal angle for -1,500
1. Add 360 to -1,500
2. Keep adding 360 to your answer until you get a positive number
3. I added 360 5 times to get a positive coterminal angle of 300

Find a negative coterminal angle for 500
1. Subtract 360 from 500
2. Keep subtracting until you get a negative number
3. I subtracted 360 2 times to get a negative coterminal angle of -220

EXAMPLE 4: Adding and Subtracting coterminal angles with 2pi

Find a positive coterminal angle for pi/4
1. Add 2pi to pi/4
2. Your answer will be 9/4 when you plug it into your calculator

Find a negative coterminal angle for pi/2
1. Subtract 2pi from pi/2
2. Your answer will be -3/2 when you plug it into your calculator

This whole section is super easy as long as you remember all over you formulas and when to add and subtract certain numbers.

This week in adv math we learned a whole lot of stuff. We started on chapter 7 and learned how to find the measure of an angles in both degrees and radians to find coterminal angles, we also learned that there are two units of measurements. Which would be degrees and radians.

I'm gunna work a problem and show you how to convert degrees to radians.Ok.

Ex. 1 convert 26 degrees into radians
1.Plug into formula 26xpi/180
2.multiply to get 26pi/180
3.now plug that into your calcalater
4.then you now get .62

I know this is crazy but now we are gunna convert radians into degrees
Example. Convert pi/6 to degrees
1. multiply pi/6 by 180/pi
2. the pis cancle out to give you 180/6
3. divide that to give you 30 degrees

And that would be what i learned this week in advanced math. The key to passing this class is to pay attention, do you homework(most important), and to do your blogs. It was nice sharing this with you kbye.

7-1 Measurement of Angles

7-1 Measurement of Angles
This week in Advanced Math we covered Chapter 7. Our first section, 7-1, taught us how to find the measure of an angle in either degrees or radians and to find coterminal angles. As we learned, there are two units of measurement: degrees and radians.

First, lets see how you would convert degrees to radians.
The formula for converting degrees to radians is (degrees x pi/180).
EXAMPLE 1: Convert 36 degrees to radians (to the nearest hundredth).
1. Plug into formula - 36 x pi/180
2. Multiply to get 36pi/180
3. Plug 36pi/180 into your calculator.
4. You round to get .63
EXAMPLE 2: Convert 270 degrees to radians (give answer in terms of pi).
1. Plug into formula - 270 x pi/180
2. Multiply to get 270pi/180
3. 270 and 180 are divisible by 90 so your answer will be 3pi/2.

Now that we've gone over degrees to radians, lets try radians to degrees.
The formula for converting radians to degrees is (radians x 180/pi).
EXAMPLE 1: Convert 2pi/3 to degrees.
1. Plug into formula – 2pi/3 x 180/pi
2. Pi cancels out.
3. Then, multiply 2/3 x 180
4. Your answer is 120 degrees.
EXAMPLE 2: Convert 2.5 x 180/pi to degrees (give answer to nearest tenth of a degree).
1. Plug into formula – 2.5 x 180/pi
2. Plug 2.5 x 180/pi into your calculator
3. Round and your answer will be 143.2 degrees

Along with these two formulas, we were also taught how to find negative and positive coterminal angles. It may sound difficult, but we all know that it’s pretty simple.
To find a positive coterminal angle, add 360. To find a negative coterminal angle, subtract 360. Let’s try that.
EXAMPLE 1: Find a positive and negative coterminal angle to -1000 degrees.
1. To find a positive coterminal angle, add 360 until your answer is positive.
2. You had to add it three times to get 80 degrees. There is your positive.
3. Now, subtract 360 from -1000.
4. Your answer should be -1360.
**** When finding coterminal angles, your answers will vary. Just make sure it’s positive when it should be and negative when it should be.


The key to understanding Chapter 7 is learning your formulas! It’s much easier to learn when you know what you’re plugging your numbers into :)

7-3 Sine and Cosine Functions

This week in Advanced Math we were introduced to Chapter 7. This chapter included angles, arcs, sectors, the sine and cosine functions, and also the tangent, secant, cotangent, and cosecant trigonometric functions. In lesson 7-3 we were taught how to use sine and cosine to find values of these functions and solve simple trigonometric equations.

-Sine of Ѳ is defined sinѲ
sin is defined in the coordinate plane as:
y → sin
r/hypotenuse → radius/ a²+b²=c²
-Cosine of Ѳ is defined cosѲ
cos is defined in the coordinate plane as:
x → cos
r/hypotenuse → radius/ a²+b²=c²

-Sin is positive in the quadrant where (y) is positive and vice versa.
-Cos is positive in the quadrant where (x) is positive and vice versa.

Example: If the terminal ray of angle α in standard position passes through (-2, 4) find sinѲ and cosѲ.
sinѲ=(-2)/(2√5) =(2√5)/(2√5)= (-4√5)/(5)

cosѲ=(-4)/(2√5) =(2√5)/(2√5)= (-8√5)/(5)

The key to Chapter 7 is learning the formulas. If you know your formulas, the process becomes much easier to work through.

The thing that tripped me up on this lesson was making sure the bottom number was rationalized. Remember to rationalize and complete each step of the process.

Reference Angles for Sine and Cosine

Finding reference angles involves a series of steps.

Finding a reference angle of sine or cosine

1. Find where sine or cosine lies on a unit circle
The unit circle is a circle with a radius of 1. The unit circle's center usually lies on the origin at point (0,0).
2. Find if sine(Y) or Cosine(X) is positive or negative at the location where you stopped in step 1.
3. Write down whether it's + or - then write the function.
Ex. - cos + sin or sin.
4. Add or subtract 180 until you have an angle with an absolute value less than 90.
5. Write down that angle's absolute value.

Examples

1. Find a reference angle for:

a) Sin 430
First we imagine a unit circle or draft one up if needed.

Next, we'll go around 430 degrees and stop in that quadrant.

We notice that the number is greater than 360 degrees so we can just subtract 360 from our number and use that angle instead.
430-360=70

So we can go around 70 degrees and stop at Quadrant I.

Sine correlates to your Y coordinates. in quadrant I your Y's are always positive. So we either write + sin or sin as the first part of our answer.

Lastly we'll subtract 180 until we reach a number whose absolute value is less than 90 and we'll write down that absolute value.
430-360=70; |70|=70

And our answer is; sin 70

b) Cos 584028394392384710

Now we have a pretty large number here, right?

The easiest way to do a problem like this, instead of subtracting adding 360 or subtracting 180 over and over is to use division.

The first thing we need it do is find where it lies on the unit circle.

We know every 360 degrees is a full circle so let's take out as many 360s as we can

584028394392384710/360

After you divide take your remainder and that's what you'll go with on your unit circle.
The remainder comes out to 311. What this means is if you subtracted 360 over and over until you reached 311. It's just a shortcut method for big numbers.

311 on your unit circle ends in Q IV. Since we're working with cosine we're working with X's. your X's are positive in the quadrant IV so your answer must be positive. So we'll write that down.
Cos =
or
+Cos =
Now to find the angle. Subtracting 180 twice is the same as subtracting 360 once. Which means we'd come to 311 eventually, as we did earlier.

The easiest way to go from there is subtract 180 until you get an absolute value less than 90.
311-180 = 131
-180= -49

-49's absolute value is less than 90 so we'll use that absolute value.

cos 49

So we know the reference angle of Cos 584028394392384710 degrees is cos 49 degrees.

7-4 Finding Reference Angles

This past week in advanced math we covered most of chapter seven. At first it seemed overwhelming, but once you look at everything just one step at a time I really isn’t that complicated.
To find a reference angle:

1. Find the quadrant the angle is in
2. Determine if the trig function is positive or negative
3. Subtract 180 degrees from the angle until absolute value of theta is between 0 and 90 degrees.
4. If it is a trig chart angle, plug it in. If not, leave it or plug it in the calculator.

***Hint-- evaluate means the answer is a number unless otherwise specified.***
**If asked for radiants-
0 degrees= 0

30 degrees = pi/6

45 degrees = pi/4

60 degrees = pi/3

90 degrees = pi/2**

Example One------
Sin 405
First, you must decide what quadrant on the graph 405 is in. Since it is bigger than 360 you must go around the grid once and a little more.
Once you have done that, you see sin is positive since it is in the first quadrant.
+ Sin 405
Next, you must subtract 405 from 180
180-405= -225
Then take the absolute value of -255
I-225I
Which equals 225
Since 225 is larger than 90 you must subtract from 180 again.
180-225= -45
Take the absolute value again
I-45I
Which gives you 45
Since 45 is between 0 and 90 you don’t have to do anything else unless it asks for radiants
If they ask for radiants, look back at the chart above and you can see that 45 = pi/4

Converting Radians and Degrees.

In the past two weeks, I caught on to a few new math formulas. The two that

I totaly understand, are the formulas for turning Radians into Degrees and turning Degrees

into Radians. Everybody relates circles to degrees; an entire circle is 360 degrees, 3/4 of a

circle is 270 degrees, half of a cirlce is 180 degrees, and 1/4 of a circle is 90 degrees. Well in

math, there are always multiple ways to read and work problems. Just as you can measure

a road by meters or inches, you can measure angles by Radians and Degrees.

Here are the names for the radians and degrees of a circle.

CIRCLE DEGREES RADIANS

4/4 360 2 pi

3/4 270 3 pi/2

2/4 180 pi

1/4 90 2/pi

Now, the formula to convert degrees to radians is:

Degree (pi / 180)

The formula to convert radians to degrees is:

Radians (180 / pi)

~~~~~~Here are some examples to help you get an idea of whats going on!~~~~~~

~~~( * will be used as degree sign)~~~

Example #1 Convert 200* into radians

~ plug 200* into the formula

~ 200*(pi/180)

~ work

~ 200/180 pi radians or 3.49 radians

Example #2 Convert 1.4 radians into degrees

~plug 1.4 into the formula

~1.4 (180/pi)

~work

~80.2*

Hopefully this will help anybody that would be in need for these formulas.

reference angles

I understood more the first week of school than I did this week. I had a lot of trouble when we started to do the word problems. It was hard to figure out which formula to use. The things I did understand were how to find a reference angle and how to convert radians to degrees and degrees to radians.
To find a reference angle, there are 4 steps that you need to follow.
1. You need to find the quadrant that the given angle is in.
2. Next, you have to figure out whether the trig function is positive or negative.
3. Then, you subtract 180 degrees from the angle until you have an absolute value of between 0 and 90 degrees.
4. If the answer is on your trig chart, plug it in. If not, then just leave it alone.

Example 1: Find the reference angle of cos 329 degrees
a. 329 degrees will end up in quadrant IV
b. The points for quadrant IV (0, -1)
c. Cos refers to x, so this trig function is positive
d. Since 329 is bigger than 90, you have to subtract 180 until it is equal or smaller.
e. Once you subtract 180, you end up with -31, so the absolute value of -31 is 31.
The answer turns out to be +cos31 degrees
Example 2: Find the reference angle of sin 331 degrees
a. 331 degrees will also end up in quadrant IV
b. The points for quadrant IV (0,-1)
c. Sin refers to y, so this trig function is negative
d. Since 331 is bigger than 90, you have to subtract 180 until it is equal or smaller.
e. Once you subtract 180, you end up with -29, so the absolute value of -29 is 29.
The answer turns out to be –sin29 degrees

** A hint is when it says to evaluate, that means you need a number answer unless it is specified otherwise.

Reference Angles

This week we learned about reference angles. Reference angles are angles that are formed by the terminal ray and the x-axis. All reference angles are acute angles. Reference angles can be formed in each quadrant. There are formulas to determine the reference angle in each quadrant.

In the first quadrant, the reference angle is always equal to theta. To find the reference angle in the second quadrant, you would do the equation, “180-theta”. In the third quadrant, you would the equation, “theta-180”. In the fourth quadrant, you would do the equation “360-theta”.

EXAMPLE PROBLEM: Find the reference angle of an angle that is 193 degrees.

1. First, determine the quadrant in which the angle is in.

2. 193 degrees is in the 3rd quadrant, so I would use the formula "theta-180".

3. 190-180 is equal to 13 degrees.

4. Draw the reference angle to x-axis.

Since theta was positioned in the x-axis, I found the reference angle by using the formula "theta-180". Once I found the reference angle, I formed the angle to the x-axis.

An important note to remember is that reference angles are not formed to the y-axis, but always to the x-axis. There are many things you can do once you find the reference angle. For example, when you draw the reference angle to the x-axis, a special right triangle is formed. From that point, you can figure out special trigonometric identities such as sin, cosine, and tangent.

Over the first two weeks of school we start chapter 7. When Mrs. Robinson first started to show us what to do i knew it was going to be hard. I was right. We began learning the beginning of trig which was converting degrees to radians and vice versa. Then we learned how to find arc lengths, radi, and thetas. The one thing i could really understand had to do with the unit circle and sine and cosine which is called "Reference Angles". Once you learned your unit circle and remembered your quadrants you were set, and not to forget how to find your hypotenuse. Although, there are pathagorean triples to help out with that. The way to do these things are:

Sin- in the coordinate plane as y/r or y/hypotenuse
Cos- in the coordiante plane as x/r or x/hypotenuse

Example - If the terminal ray of an angle from the origin passes through (-4,5) find sin(theta) and cos(theta).

*To find either one you would first find your two points and make the triangle. To find your 3 point (hypotenuse) you use the pathagorean thereom which is a^2 + b^2 = c^2

*For sin find the x, which in this case would be 5. So, 5 would go over the hypotenuse which is Squareroot-41. By the rules you know you can never have a squareroot at the bottom. To get rid of it you multiply both the top and the bottom by the squareroot (41). The ending answer would be:

5Squareroot41/41

*For sin find the y, which would be -4. You would follow the same process. By the end your answer would be:

-4Squareroot41/41

The unit circle has to do with the degrees 90,180,270 and 360. The points for each are:
90degrees - (1,0) Trig Chart: Pi/2 = 90degrees
180degrees - (0,1) Pi = 180degrees
270degrees - (-1,0) 3pi/2 = 270degrees
360degrees - (0,-1) 2pi = 360degrees

--The unit circle has a radius of 1.

Example - sin 270degrees = y/r = 0/1 = 0

Example - cos450degrees ; In order to find the point in the unit circle the degrees must be less than 360. To get lower subtract 360 until number is found.

450-260=90;cos90degrees = x/r = 0/1 = 0

--If your degree is negative, you will go around the circle backwards. For example: cos-95. Instead of being in quadrant two you will be in quadrant three because you go backwards from 360.

Finding reference angle goes off of what we learned before. To do this the rules are:

1. Find the quadrant the angle is in
2. Determine if the trig function is postive or negative.
3. subtract 180degrees from the angle until theta is bewteen 0 and 90 degrees.
4.If it is a trig chart angle plug in

Example - Find the reference angle for sin465.

*Subtract 360 = 105. 105degrees is in quadrant 2. Sin is postive in this quadrant. Subtract 180 from 105

+sin(-75)

*This answer will never be negative because you take the absolute value.

= +sin75degrees

Example - Find the reference angle of sin-45

*Because 45 is already lower than 360 there is no need to subtract. -45 is in quadrant 4 and sin is also negative. Because -45 is also less than 90 there is no need to subtract 180. Take the absolute value which equals

-sin45

*45 is in the trig chart which equals squareroot2/2

This is only a breif explanation of reference angles, but from what we learned there are many other problems and solutions that can be ran across.

I had more trouble this week then i did last week. Last week we went over converting degrees to radians and radians to degrees. I found this very easy. For instance, to convert from degrees to radians you simply multiply by pie over 180. Converting radians to degrees you multiply 180 over pie. When we learned how to do word problems I found it difficult at first until I learned my formulas. Reference angles were fairly easy once you figured out which quadrant it was in. You would simply determine whether it was positive or negative and then either subtract 360 or 180.

Example 1) The moon is about 4x10^5km from Earth and its apparent size is about 0.0087 radians. What is the moon's approximate diameter?
-for this problem you must first determine how far the Earth is from the moon, its apparent size, and the approximate diameter.
theta=0.0087 radius=4x10^5
s=?
*first choose which formula would be best to work this problem.
s=rtheta
k=1/2r^2theta
k=1/2rs
*s=rtheta would be the best formula to use.
*set it up s=(4x10^5)(0.0087)
*multiply (4x10^5) and (0.0087)
-when plugging it into your calculator it should be 4xE5 x 0.0087
*simply multiply both numbers together
**your answer should be s=3480km

I pretty much understood everything we went over last week. But i'm having trouble with a few things from this past week. I'm still a little iffy on the 5 trig function. I'm not sure how you find the hypotenuse. Inverse of angle is still confusing be a bit too.

How to find the sector of the circle

How to find the sector of the circle

Formulas:
1. s=(r)(theta)
(s= arch length) (r= radius) (theta= central angle)

2. k= (1/2)(r^2)(s)
(k= area of the sector)

3. k= (1/2)(r)(s)

Next here are your steps:
1. find out what is given.. for example s=? r=? k=? and theta=?
You must find at least two of these in order to figure out this problem.

2. you are going to want to make sure that theta is in radians.
if theta is in degrees then you need to multiply that by (pie/180) to convert degrees into radians.

3. next, you are going to need to choose one of the formulas to plug into.
you want to choose the formula that works best with the problem.
for example if you have s= 4 cm and k= 36 cm you are not going to want to choose formula #1.
you would want to choose formula #3.

4. then, after you find one of the missing variables you will have one more variable still to find.
so, you are going to plug into another equation that fits the found variables and plug into the appropriate equation.

Here is an EXAMPLE:
A sector of a circle has an arc length of 4 cm and an area of 52 cm^2. Find its radius and the measure of its central angle.

So now follow the steps..
What are the given variables?
s= 4cm k=52cm^2 r=? theta=?

Now you must use the appropriate formula
formula #3 will work the best. [k=(1/2)(r)(s)]
52cm^2=(1/2)(r)(4)
52cm^2=2r
r= 26cm

Now that you found the answer for the variable r you can plug into another formula to find theta.
the best fit formula to find theta is formula #1 [s=(r)(theta)]
4cm= (26cm)(theta)
theta=(1/6 radians)

This is how you find the sector of the circle.

Reference Angles

This week in advanced math we covered most of chapter seven. We learned how to convert degrees to radians and vice versa. In seven-two we learned we learned some important formulas that we will need. We also learned how to read word problems and pull certain parts of the formula out of the word problem. Next we learned about sin, cos, the unit circle , and pythagorean triples. Cos always has to do with x and sin always has to do with y. We went on to learn the Trig chart, tan, cot, sec, and csc. But what i found easiest to do for me was finding reference angles.

To find a reference angle you must first find the quadrant that the angle is in. Next you must determine if the trig function is positive or negative. Then you subtract 180 degrees from the angle until the absolute value of theta is between 0 and 90 degrees. Last of all if the angle you end up with is a trig chart angle you plug in the trig chart number. If not you either leave it as it is or plug it into your calculator.

Ex. Find the reference angle for cos 525 degrees

First you find which quadrant 525 degrees is in on the graph to determine if cos is positive or negative.

525 degrees is located in quadrant 2 which has a x value of -1, which means cos(x/r) equals -1/1, therefore cosine is negative.

Next continue to subtract 525 by 180 until you get in between 0 and 90 degrees.

525-180= 345-180=165-180=-15

The absolute value of -15 =15

Therefore -cos=15 degrees

Test

Test post

Pythagoren Triples

This week we continued Trigonometric Functions in chapter seven. Chapter seven discussed angle measurement, circle sectors, and trigonometric functions. We learned about the functions of sine, cosine, and tangent. We also learned about the unit circle, and its relationship to the “Trig Chart.” The trig chart is a chart used to show functions of special angles. Both the unit circle and the trig chart are important, so it was necessary that we learn and remember both of them. We also learned about Pythagorean triples and how they would help us.

Pythagorean Triples saves the time it would take you to use the Pythagorean Theorem.

Ex.

What is a Pythagorean Triple?

- A set of numbers that form a right triangle.

- m and n will form a Pythagorean triple.

Suppose :

- m and n are positive.

- m <>

- n^2-m^2 , 2mn, and n^2+m^2 = a Pythagorean Triple.

The smallest triple is (3,4,5)

m=1 n=2

1 is positive. 2 is positive .

1 is less than 2 .

- 2^2-1^2= 4-1= 3 the first number is 3.

- 2(1)(2) = 4 The second number is 4.

- 2^2+1^2= 4+1=5 The last number is 5.

Therefore, ( 3,4,5) is a Pythagorean triple.

Tester

:)

Tester :D

Tester.

It worked. =)

Test!

Test blog.

Trevor

hey this is trevor

test

Tester blog