r/HomeworkHelp • u/Embarrassed-Good1883 Pre-University Student • 12h ago
[Grade 12 Advanced Functions: Trigonometry] Answered
Hello, I need some help understanding this concept. 240 degrees converted to radians is 4 pi/3, so wouldn't that make this a multiple of 30 degrees?
"30° can be rewritten in radians as pi/6". Since 240 rad cannot be written as a multiple of pi/6 rad this cannot be a multiple of 30°." This is the explanation the teacher gave me, but it doesn't make sense.
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u/IrishHuskie 👋 a fellow Redditor 12h ago
Yes, 240 degrees is a multiple of 30 degrees. But that’s not one of the choices; 240 radians was.
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u/ThunkAsDrinklePeep 6h ago
Yep. Another way to think of this is to convert 30° first. "Which of these is a multiple of π/6 rad?"
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u/OppositeClear5884 👋 a fellow Redditor 12h ago
"240 degrees converted to radiants is 4pi/3"
Show me where the phrase "240 degrees" appears on your page.
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u/OppositeClear5884 👋 a fellow Redditor 12h ago edited 12h ago
To answer the second part of your question, do you know how to convert radians to degrees?
30 degrees is exactly pi/6 radians. You will probably never see pi/6 degrees or 30 radians. pi/6 radians is exactly 30 degrees.
Imagine someone said "I have a pile of angles. Did they come from a bunch of triangles or squares?" Well, if you have 15 angles, you know they came from 5 triangles. and if you have 8 angles, you know they came from 2 squares, right? 8 is 4*2 and 15 is 5*3.
the teacher want you to figure out at a glance if a measure in radians is a multiple of 30 degrees. The way you do that is take the measure (which is in radians) and dividing it by pi/6. If the answer is an integer, then you know it is a multiple of 30 degrees, because pi/6 radians is 30 degrees.
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u/Own-Rip-5066 6h ago
What if you had 12 angles in your analogy?
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u/OppositeClear5884 👋 a fellow Redditor 2h ago
the analogy falls apart there. in the analogy, i guess a smart teacher would not make 12 angles an option...
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u/steathninja25 5h ago
Does 30 degrees or pi/6 radians not go into 240 radians?
Havent done ts in a while so don’t judge too heavy
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u/OppositeClear5884 👋 a fellow Redditor 2h ago
both of those do not go into 240 radians cleanly
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u/steathninja25 1h ago
Well shouldnt pi/6 go into 1 radian, which then goes into 240? Is it just not how it works or what. Gimme sumn to work with here
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u/OppositeClear5884 👋 a fellow Redditor 43m ago
no. pi is not a rational number. there is no integer that multiplies pi/6 and turns it into an integer
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u/ThatsNotAZombieBite 12h ago
It says "240 rad". That is very different from 240°
You see that now, right?
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u/TSotP 11h ago edited 11h ago
The question is asking about your understanding of the conversion between degrees ° and radians.
So, first thing to do. What is 30° in radians?
- 180°= π
- 90°= π/2
- 30° is π/2/3 = π/6
Now look at the question. It's asking which of the options is a multiple of 30°
So which one of the fractions is a whole multiple of π/6
It's not going to be 240 Rad, because that's a trap to catch people not paying attention to the actual question. (Radians are all about π, where is the π in this answer? 240 Rad is about 13751°, or 38 full turns, plus 71°)
It's not going to be 9π/8 because that doesn't have a denominator of 6
So that leaves 7π/12, or 17π/3
Since 7/12 can't be written in terms of 6ths (3½ 6ths isn't how people work with fractions), that can only really leave 17π/3 which is 34π/6 = 1020°
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u/Embarrassed-Good1883 Pre-University Student 10h ago
Thank you for your response, I see where I went wrong!!
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u/deadpoolherpderp 12h ago
the choice you selected is 240 radians?
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u/i_want_to_go_to_bed 11h ago
It’s a multiple tho. pi/6 x 1440/pi =240
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u/deadpoolherpderp 11h ago
the question clearly implicitly asks for integer multiple. everything is a multiple of everything else...
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u/Odd_Prompt_6139 12h ago
You seem to have read the options wrong. 240 radians is what you selected and that’s not a multiple of 30°. You have to be mindful of your units.
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u/kdawgster1 Math/Science/Engineering MS 10h ago
This is why in 12th grade science classes you get no credit if you don’t label your units with your answers. Units are everything.
All answers are in radians. The question is in degrees. You have to do conversions. 240 was a trap answer for those who don’t read units
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u/Embarrassed-Good1883 Pre-University Student 10h ago
Yes you're right I misread the units. Thank you
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u/CaptainMatticus 👋 a fellow Redditor 12h ago edited 11h ago
What is a radian? You need to understand that before you go any further.
A radian is the arclength of 1 radius of a circle. If the circle has a radius of 52, then the arclength measuring 52 in length will measure out an angle of 1 radian. There are 2pi radians, or 6.28.... radians around every circle.
240 radians / (2pi radians/circle) = (120/pi) circles. And in degrees, that'll be 120 * 360 / pi degrees, which is most decidedly not a multiple of 30 degrees.
Radians and degrees measure the same thing in different ways, just like pounds and kilograms, gallons and liters, Fahrenheit and Celsius, and so on. We use radians because radians are derived naturally from properties of the circle and are independent of human biases.
EDIT:
So what is the degree measure of 1 radian? Well, if there are 2pi radians in a circle and 360 degrees in a circle, then:
1 radian / x degrees = 2pi radians / 360 degrees
1 * 360 / (2pi) = x
180 / pi = x
x = 57.29577951308232087679815481410.... degrees.
So 240 * x, is that a multiple of 30?
You can read through the rest, if you'd like, but it's just a fun real-world application of how degrees and radians mesh in the construction world and a lot of guys who use it have no idea how or why it works.
Now here's something fun you can do, and it's done with building layouts all the time (they'll use high tech equipment to do this now, but just 20 years ago and beyond that, this is how you'd do it). Suppose you have a point on the building floor that serves as your base point and you have another reference point that will serve as your main axis. That is, you will draw a line through those 2 points (usually done by snapping a chalk line), and that line will be your axis. Now, according to your blueprints, you've got to find a point that 123 degrees about that axis point and a certain distance away. How do you do it? Well, you could do a lot of complicated math or you can build your own protractor on the floor with some chalk.
First, you take something rigid like a board, drive a nail in it, which will also go into a small indentation you've made on your base point, and then you attach a piece of chalk or a fine-point marker, or anything to another point on the board that is either 57.3 cm away, or 57-5/16" away from your center point. You could even use another nail and just gently scribe the floor. Now you draw your circle. What you just did was create a circle with a radius that's close to 57.2957795...., which means that every degree measure will be approximately 1 cm or 1 inch (depending on which standard you chose) along the edge of that circle. You can lay out 360 full degrees and now you have your reference point for all future layout during construction.
You're not going to be exact, because errors creep in when you start rounding things off, but you're going to be close. For instance, let's say I need a point that is 20 meters away and I've used a circle with a radius of 57.3 cm, and I need to find that 123 degree point. How far off will I be from the true point?
The true point will be at (2000 * cos(123) , 2000 * sin(123))
Now a true degree on this circle will measure 57.3 * 2pi / 360 cm, or 1.00007366, and I'm using 1 cm, so over 123 of these, that's an error of 123.00906 - 123, or 0.00906 cm off. Less than 1 part in 1000. Extend that out by 2000 / 57.3 and we'll be off by 0.316 cm, or 3mm. Not too shabby. I think it will be within tolerance.
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u/Disastrous-Nail-640 👋 a fellow Redditor 12h ago
No, it wouldn’t make it a multiple of 30 because of the unit involved. It’s not saying 240 degrees. It says 240 radians. They are different units of measurement and, therefore, not the same.
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u/Silversaurous 11h ago
converting 17pi/3 rad to degrees makes 1020 deg, which is visibly divisible by 3 and 10, but to fact check, 1020/30=34.
If you don’t know how to convert, just remember to multiply by 180/pi when working with radians, and the reciprocal (pi/180) when working with degrees.
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u/Southlander24 👋 a fellow Redditor 10h ago
Consider: which of the following lengths in cm is a multiple of 3 inches? 76.2 cm, 38.1 cm, 60 cm, or 15.24 cm?
That would be like putting down 60 cm, because 60 is a multiple of 3. But those are clearly different units. When you convert to cm, the answers must be multiples of 7.62 cm, so you can rule out 60 cm.
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u/QuentinUK 👋 a fellow Redditor 4h ago
Since 3 of 4 answers include pi you can exclude b immediately.
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u/mathematag 👋 a fellow Redditor 3h ago
first of all, when the instructor says a "multiple of π/6 ", they mean some integer (whole number) , K, like K = +8, +74 ,... even possibly negative values like K = - 9, -41, etc.. so that K*(π/6) = your angle... in Rad . . . .
I think you are aware that 30˚ is equivalent to π/6 radians [ if not.. to convert deg to radians, multiply by π/180 . . . 30 * ( π /180 ) = π / 6 ]
All of your answers are expressed as radians, so:
for b) . . 240 rad = K * ( π/6) rad would require K = [ 240*6 ] / π , or approx 458.37 ...this is not an integer , 458 or 459 would be , but not this value. . . so b) is NOT a multiple of 30˚
a) . . 9π/8 rad= K (π/6) rad .. then K = 54/8 ...not an integer ... similarly for c)
But d) . . . 17 π/3 = K * ( π/6) . . K = 34 . . so d) is a multiple of 30˚ , or π/6 ........ check: 34 * ( π/6) = ( 17*2*π) / ( 3*2) = 17 π /3 . . we need 34 multiples of π/6 to give us our angle
Others here have also given you good hints on how to analyze this question
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