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Trigonometry Misunderstanding between converting radians and degrees

Casio

Member
Feb 11, 2012
86
OK I have a right angled triangle and one angle is 1/8 pi.

So I know two angles then, 90 degrees and 1/8 pi.

I also know the hypotenuse has a length 5 units.

I am required to find the angle remaining in radians.

The problem I have is that although I can do conversions on the calculator they are always decimal notation, so radians of the form pi / 6 as example cannot be completed.

So really my question becomes;

if I have 1/8 pi = 390 and I know the other angle is 900 , how do I convert 890 to radians?

pi / 180 x 89 = 1.55 radians, but this is not in the correct format like,

pi/2 which would represent 900

Could somebody please advise how I change decimal notation over to this format please.

Kind regards

Casio (Happy)
 

SuperSonic4

Well-known member
MHB Math Helper
Mar 1, 2012
249
OK I have a right angled triangle and one angle is 1/8 pi.

So I know two angles then, 90 degrees and 1/8 pi.

I also know the hypotenuse has a length 5 units.

I am required to find the angle remaining in radians.

The problem I have is that although I can do conversions on the calculator they are always decimal notation, so radians of the form pi / 6 as example cannot be completed.

So really my question becomes;

if I have 1/8 pi = 390 and I know the other angle is 900 , how do I convert 890 to radians?

pi / 180 x 89 = 1.55 radians, but this is not in the correct format like,

pi/2 which would represent 900

Could somebody please advise how I change decimal notation over to this format please.

Kind regards

Casio (Happy)
The angles of a triangle must always sum to 180 degrees or $\pi$ radians.

Because of the way the radian is defined we know that $90^{\circ} = \frac{\pi}{2} \text{ rad}$ (although normally you'd not put "rad" down since radian is the de-facto standard for measuring angles)

1 radian is defined as the angle subtended by an arc with length equal to the radius.

That means the total number of radians in a full circle is the amount of radii in the circumference which is $\frac{2\pi r}{r} = 2\pi$

In turn this is equal to 360 degrees and if we divide both sides by two we arrive at $180^{\circ} = \pi $ and by extensions $90^{\circ} = \frac{\pi}{2}$

The former is an important conversion factor and will help you a lot if you can recall/derive it

That leaves you with the equation: $\frac{\pi}{2} + \frac{\pi}{8} + \theta = \pi$

edit: where $\theta$ is the angle you're trying to find which will be in radians. Leave your answer in terms of $\pi$ (unless explicitly told to round off to a set number of decimal points/significant figures)
 

Casio

Member
Feb 11, 2012
86
The angles of a triangle must always sum to 180 degrees or $\pi$ radians.

Because of the way the radian is defined we know that $90^{\circ} = \frac{\pi}{2} \text{ rad}$ (although normally you'd not put "rad" down since radian is the de-facto standard for measuring angles)

1 radian is defined as the angle subtended by an arc with length equal to the radius.

That means the total number of radians in a full circle is the amount of radii in the circumference which is $\frac{2\pi r}{r} = 2\pi$

In turn this is equal to 360 degrees and if we divide both sides by two we arrive at $180^{\circ} = \pi $ and by extensions $90^{\circ} = \frac{\pi}{2}$

The former is an important conversion factor and will help you a lot if you can recall/derive it

That leaves you with the equation: $\frac{\pi}{2} + \frac{\pi}{8} + \theta = \pi$

edit: where $\theta$ is the angle you're trying to find which will be in radians. Leave your answer in terms of $\pi$ (unless explicitly told to round off to a set number of decimal points/significant figures)
Thank you for the above effort you have put in to explain that, it is very much appreciated, but may I ask;

if

pi/2 = 900
pi = 1800
2pi = 3600

Then how do I convert 890 to radians like the left hand side of the equals signs above?

This is where my misunderstanding lies, I can convert to and from degrees to radians and vica/versa, but can't get the understanding how I would change decimal to radians like;

2700 = 3pi/2

The question becomes how do I do the above?

Kind regards

Casio
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,043
It seems like you get the basic conversion idea, as you said.

\(\displaystyle 89 ^\circ = 89 \times \frac{\pi}{180}\) radians. If you input "89/180" on a calculator you most likely will get a decimal answer of \(\displaystyle .4944 \pi\) radians or just \(\displaystyle 1.55\) radians.

As others have said, normally don't convert $\pi$ to a decimal so the \(\displaystyle .4944 \pi\) form is better. Now I think you're asking about how to make this decimal a fraction, correct?

Well if you're using a calculator you can input something like "89/180 F->D" or "89/180 D->F". There are usually buttons to force the output to be a fraction or force it to be a decimal, as well as switch between the forms. That depends on your calculator.

If doing it by hand then you would need to try to find a common factor for both and simplify. It just so happens that 89 is prime, so this can't be simplified anymore therefore you could write you final answer as \(\displaystyle \frac{89 \pi}{180}\)
 

Reckoner

Member
Jun 16, 2012
45
if I have 1/8 pi = 390 and I know the other angle is 900 , how do I convert 890 to radians?
I'm wondering where you got the \(89^\circ\) from. Could you explain that?
 

Casio

Member
Feb 11, 2012
86
The angles of a triangle must always sum to 180 degrees or $\pi$ radians.

Because of the way the radian is defined we know that $90^{\circ} = \frac{\pi}{2} \text{ rad}$ (although normally you'd not put "rad" down since radian is the de-facto standard for measuring angles)

1 radian is defined as the angle subtended by an arc with length equal to the radius.

That means the total number of radians in a full circle is the amount of radii in the circumference which is $\frac{2\pi r}{r} = 2\pi$

In turn this is equal to 360 degrees and if we divide both sides by two we arrive at $180^{\circ} = \pi $ and by extensions $90^{\circ} = \frac{\pi}{2}$

The former is an important conversion factor and will help you a lot if you can recall/derive it

That leaves you with the equation: $\frac{\pi}{2} + \frac{\pi}{8} + \theta = \pi$

edit: where $\theta$ is the angle you're trying to find which will be in radians. Leave your answer in terms of $\pi$ (unless explicitly told to round off to a set number of decimal points/significant figures)
I am still confused with the conversions?

pi/2 + pi/8 + theta = pi

That I understand, but

pi/2 = 1.57 radians

pi/8 = 0.39 radians

pi = 1800

180 / pi x 1.18 = 67.610

I still can't get this angle in radians, where I know the solution is 3pi/8, but can't convert to it?

Regards

Casio(Smile)
 

Casio

Member
Feb 11, 2012
86
I'm wondering where you got the \(89^\circ\) from. Could you explain that?
Yes I can, I have missunderstood the interpretation of the result given by the calculator.

Sorry

Casio
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,043
I am still confused with the conversions?

pi/2 + pi/8 + theta = pi

That I understand, but

pi/2 = 1.57 radians

pi/8 = 0.39 radians

pi = 1800

180 / pi x 1.18 = 67.610

I still can't get this angle in radians, where I know the solution is 3pi/8, but can't convert to it?
Where is the 1.18 coming from?

You start with \(\displaystyle \frac{\pi}{2} + \frac{\pi}{8} + \theta = \pi\) and find theta by subtraction.

\(\displaystyle \theta = \pi - \frac{\pi}{2} - \frac{\pi}{8}\)

Get everything in terms of $\pi$ and you're done, I believe. There's no need to convert things to decimals.
 

Reckoner

Member
Jun 16, 2012
45
180 / pi x 1.18 = 67.610

I still can't get this angle in radians, where I know the solution is 3pi/8, but can't convert to it?
The 1.18 is already in radians. If you want to get an exact answer, you should avoid converting \(\pi\) to a decimal approximation. Really, it isn't even necessary to do any conversion.

SuperSonic4 posted the equation that you need to solve:

\[\frac\pi2 + \frac\pi8 + \theta = \pi\]

Solving for \(\theta\) gives

\[\theta = \pi - \frac\pi2 - \frac\pi8.\]

Do you know how to add and subtract fractions together? You will need to make a common denominator in these three fractions in order to combine them.
 

Casio

Member
Feb 11, 2012
86
The 1.18 is already in radians. If you want to get an exact answer, you should avoid converting \(\pi\) to a decimal approximation. Really, it isn't even necessary to do any conversion.

SuperSonic4 posted the equation that you need to solve:

\[\frac\pi2 + \frac\pi8 + \theta = \pi\]

Solving for \(\theta\) gives

\[\theta = \pi - \frac\pi2 - \frac\pi8.\]

Do you know how to add and subtract fractions together? You will need to make a common denominator in these three fractions in order to combine them.
I can find denominators in normal fractions but am not sure when pi is involved?

The other thing is if the answer to the question is 3pi / 8

which is 1.18 radians, why change the denominators?

Regards

Casio
 

SuperSonic4

Well-known member
MHB Math Helper
Mar 1, 2012
249
I can find denominators in normal fractions but am not sure when pi is involved?

The other thing is if the answer to the question is 3pi / 8

which is 1.18 radians, why change the denominators?

Regards

Casio
The answer is $\dfrac{3\pi}{8}$ but leave it in that form because it's the exact form (have you covered the exact solution to equations? In short it's leaving surds and other irrational numbers like pi in the final answer instead of rounding them off)

Changing the denominator is standard procedure for fractions. Indeed, since $\pi$ is in the numerator it doesn't change how you deal with the denominator because they're rational. If you want to deal with rational numbers divide through by $\pi$ to get $\dfrac{\theta}{\pi} = 1 - \dfrac{1}{2} - \dfrac{1}{8}$. Just don't forget to multiply by $\pi$ at the end to get the answer.

------------

edit: In this example 1.18 is only an approximation of $\frac{3\pi}{8}$. Indeed if I were to say that it was 1.178097245 then it's still an approximation because $\pi$ cannot be expressed as a decimal (because it's irrational)
 

Casio

Member
Feb 11, 2012
86
The answer is $\dfrac{3\pi}{8}$ but leave it in that form because it's the exact form (have you covered the exact solution to equations? In short it's leaving surds and other irrational numbers like pi in the final answer instead of rounding them off)

Changing the denominator is standard procedure for fractions. Indeed, since $\pi$ is in the numerator it doesn't change how you deal with the denominator because they're rational. If you want to deal with rational numbers divide through by $\pi$ to get $\dfrac{\theta}{\pi} = 1 - \dfrac{1}{2} - \dfrac{1}{8}$. Just don't forget to multiply by $\pi$ at the end to get the answer.

------------

edit: In this example 1.18 is only an approximation of $\frac{3\pi}{8}$. Indeed if I were to say that it was 1.178097245 then it's still an approximation because $\pi$ cannot be expressed as a decimal (because it's irrational)
This is my problem I don't know how to convert the answers like 1.18 radians to the exact answer 3pi/8

There is nothing in the coursework that gives information or examples how to do it, so when I see the answer I am suppose to be working to I don't understand how they got there?

1.18 = 3pi/8

I can work back 3pi/8, but how do you work forward 1.18 = 3pi/8?

Casio
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
Casio, the important thing about this is that you shouldn't convert them to decimal numbers until you're done. This is probably what's confusing you.

Operate with them until you get the final answer, then, if so you desire, convert to a decimal approximation (be conscious that this hurts more than helps: if you are given a number you cannot know what calculations were done to get to it).

You were given a right triangle with one angle equal to $\frac{\pi}{8}$. Since the angles in a triangle have to sum up to $\pi$, you have the equation $\frac{\pi}{2} + \frac{\pi}{8} + \theta = \pi$, where $\theta$ is the angle you want to find. Resolving this, you get that $\theta = \frac{3 \pi}{8}$.

Leave the answer as it is!
 

SuperSonic4

Well-known member
MHB Math Helper
Mar 1, 2012
249
This is my problem I don't know how to convert the answers like 1.18 radians to the exact answer 3pi/8

There is nothing in the coursework that gives information or examples how to do it, so when I see the answer I am suppose to be working to I don't understand how they got there?

1.18 = 3pi/8

I can work back 3pi/8, but how do you work forward 1.18 = 3pi/8?

Casio
Divide by (an approximation of) $\pi$.

Let's say that $1.18 = k \pi \text{ }$ where $k$ is some constant (3/8 is what we hope to find). Since $\pi \approx 3.14$ then it follows that $1.18 = 3.14k \rightarrow k = \frac{1.18}{3.14} = 0.3758 \approx \frac{3}{8}$

It may take some estimation/knowledge of fraction-decimal values to find the right fraction. In this case to know that $0.3758 \approx \frac{3}{8}$


In my experience I find it easier to convert my angles to be in terms of pi.


OP said:
The problem I have is that although I can do conversions on the calculator they are always decimal notation, so radians of the form pi / 6 as example cannot be completed.
What calculator are you using? Most modern ones will leave the answer in terms of $\pi$ where appropriate. Of course all of this is moot if your coursework accepts decimal approximations for angles :)
 

Casio

Member
Feb 11, 2012
86
Divide by (an approximation of) $\pi$.

Let's say that $1.18 = k \pi \text{ }$ where $k$ is some constant (3/8 is what we hope to find). Since $\pi \approx 3.14$ then it follows that $1.18 = 3.14k \rightarrow k = \frac{1.18}{3.14} = 0.3758 \approx \frac{3}{8}$

It may take some estimation/knowledge of fraction-decimal values to find the right fraction. In this case to know that $0.3758 \approx \frac{3}{8}$


In my experience I find it easier to convert my angles to be in terms of pi.




What calculator are you using? Most modern ones will leave the answer in terms of $\pi$ where appropriate. Of course all of this is moot if your coursework accepts decimal approximations for angles :)
My calculator is a Texas Instruments TI 30XA, it does have a F<>D key but will not to the best of my knowledge give an exact answer because it only displays decimal answers.

Casio
 

Casio

Member
Feb 11, 2012
86
In these types of questions where the answer is known or given, can it not be answered like this;

pi - 1/8pi - pi/2 = 3pi/8

where the common denominator would be 8

Casio
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
I'm not sure what you mean, but what we all said is that you should answer like this: $\frac{3 \pi}{8}$ .
 

Casio

Member
Feb 11, 2012
86
I'm not sure what you mean, but what we all said is that you should answer like this: $\frac{3 \pi}{8}$ .
I am not exactly sure whether my method is correct either, but this is what I thought;

I was told that I had a right angle triangle, with a angle that measured 1/8pi.

I was also told that the hypotenuse was 5cm.

So I know that a triangle adds up to 1800 = pi

I also know from given information that another angle is 1/8pi,and in any right angled triangle there is a 900, so

180 - 90 - 23 = 670

so by that approximation above I know what I am looking for.

pi - 1/8pi - pi/2 = pi/8 - 1/8pi - pi/8 = 3pi/8

All I have basically done is say that 2 from 8 = 4 and 2 x 4 = 8, therefore used 8 as a denominator for pi throughout.

Thus I end up with 3 lots of pi over a denominator of 8

I.E. 3pi / 8

Casio
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
Your method is correct. I'd just avoid converting to degrees, get used to thinking in radians. (Nod)
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,043
Your method is correct. I'd just avoid converting to degrees, get used to thinking in radians. (Nod)
+1

If you always work with fractions then you won't have to worry about going from decimals to fractions. Also, the 1.18 you used was a rounded approximation so that will never lead back to the original fraction. If you keep doing more and more of these you'll get the hang of it.

Just remember:

1) Don't use a decimal approximation of $\pi$ when working with radians unless told otherwise. Almost all trig problems I've seen with radians keep everything in terms of $\pi$.

2) When adding/subtracting angles keep everything as a fraction if possible.

3) Make sure all of your angles are written in either degrees or radians, but don't use both unless there's a strange reason to.

I think this thread has run its course but I'll leave it open for a bit in case I've missed something.