The simplest, most straightforward, means of solving is to multiply the hypotenuse (12) by sin(17), multiply the hypotenuse by cos(17), then multiply the aforementioned results together, and finally divide by 2. The result is approximately 20, which is what many of the other comments herein had as an answer.
Like others, I'm surprised you went with Pythagoras for this. If you can do 12 sin 17° then you can do 12 cos 17° just as easily, and now you have base and height. So the area is: ½ × (12 sin 17°) × (12 cos 17°) = 72(sin 17°)(cos 17°) From the thumbnail, I thought the point of this video was going to be too teach the trig identity (which i had to look up just now) that (sin X)(cos X) = ½ sin (2X), so here we're just doing 36 × sin 34°. That seems a much cleaner way to get to the answer.
Due to your round off error, in an application of trig in a real world machine shop, your answer is off by 0.004 thousandths and by 0.0039 ten thousandths depending on the called out +/- tolerance, which is about the thinkness of a piece of copy paper or average human hair. It all depends on round off error
Why couldn’t I have had a math teacher like you when I was in School to have the patience as you do and be able to dumb it down to as to absorb so easily. Thanks
Like others have said why not use the cosine to find the other side Unless the point was to teach the pythagorean theory. I think it's better to stick to one topic
This is easy. Everyone knows that sin theta = opposite side/hypoteneuse. Since theta = 17 dgrees, sin17 = opposite/ 12. Opposite side = sin 17 degrees * 12 = 3.508; cos theta = adjacent / hypotoneuse. cos17 = adjacent/12. adjacent side = cos17 *12 =5.8. So, area of triangle = .5*b*h. So, area of this triangle = .5 *cos17*sin17*(12)^2 =20.17 units^2
I’m a penn state fan but I have no idea what you are talking 4:00 About. I got a phd in English and I build violins using simple geometry. Everyone doesn’t know this formula. Ha haha. But I hope The football coaches can compute.
A trig question: Sin 17° = a/12 a = 12(Sin17) = 12(0.2924) = 3.5085 b =? tan17° = a/b b = a/tan17 = 3.5085/(0 3057) = 11.4758 Area triangle: A = (1/2)b×h Here b = b = 11.4758 h = a = 3.5085 A=(1/2)(11.4758)(3.5085) =(1/2)(40.2628) =20.1314 units^2 Verify... 11.4758^2+3.5084^2=?17^2 131.6940+12.3089=?289 144.0029=❌️289❌️ ?? = 12^2. Not. =17^2
The way I did ot is system of equation which is 12=square root of x²+y² then inverse tan(x/y)=17 this is a picture of a 12 radius circle with a line through it in which where the line touchs the edge of the circle the missing sides are the x and y that make the line intersection. This came to me naturally in thought, a interesting way to find the sides of a right angle with hypotnuse and one angle using algebra. In which you do a simpe area formula 1/2×11.47×3.508=20.1309.....
First thing first, from the image, we can elicit all the angles measuse as it is a right triangle so there must be one 90 degree angle which is marked in the right bottom, then addressed 17degree angle. From prior knowledge we know, triangle consist of 180 degree, so the top angle should be 180-(90+17)=73 degrees. Now, apply soh-cah-toa over this right triangle to find out the side lengths first then, we will conclude it to the area of this rectangle. apply soh at 17 degree angle, => sin 17 = x(which is vertical side length or height)/12(hypotenuse) =>12 sin 17 = x => x = 3.508 which is similar to 3.51 its time to find out the base value as we don't know, apply 'toa' criterion, tan 73 = y(base length)/3.51 =>y = 3.51 tan 73 => y = 11.48 As we find out base and height of this triangle, so now we can place it in the triangle area formula, (1/2)*x*y =0.5*3.51*11.48 =20.15
If you are going to use the sine of 17 degrees to find the opposite side, why not just use the cosine to find the adjacent side and you don't have to do all the squaring and rooting?
I cheated. I did a quick Pythagorean based upon 144 no calculator. 11.5 X 3.5 worked,, so my guesstimate was 20 square inches area. I LIKED your explanation of sin. A LOT.
@@chrisdissanayake6979 Something like that.. You have the idea. Like I said,, I cheated. Some much too quick mental math and the first answer I came up with was actually 40.17,, but then I remembered divide by 2 Carpenter. For new house layouts, squaring with corner to corner measurements,, or to check a bath for squareness before cutting tile,, almost second nature for me. I DO almost always pull out a calculator to double check my mental math before I start cutting and nailing,, cowardice. I remember the carpenter phase, "I've cut it three times and it is still too short." Also,, when doing boringly repetitive work,, rolling out sod on new construction, or nailing shingles on a roof.. ??? You continually chant Green side up. Green side up,, Gree,,, Confession,, I have laid a roll of sod brown side up one time,, hot,, tired to stupidity,, Did not notice till I came back with the next roll and had to think about why it looked odd.
It is incorrect to say "20 units squared" vs "20 square units" - the two are not the same. If you have a square that is 8 inches on a side, what is its area? It is "64 square inches", not "64 inches squared". Semantics matter, sometimes.
@@chrisdissanayake6979as are in², cm², mi². Where we are getting confused is this: consider a room which is a square with all sides 8 feet long. We can state the room size as 8'×8', or we might say 8 feet square. Saying 8 feet squared is not technically correct, but is all too commonly used, even though feet squared is a measure of area.
@@chrisdissanayake6979 you are correct. The post you replied to is in error. He is confusing "unit square" with "unit squared". A room 8 feet square is 8x8. A room of 64 square feet, or 64 feet squared, could also be 8x8, or 16x4, etc. Because unit squared and unit square are pronounced similarly, it is preferable to say square units.
@@richardhole8429 I am not in error. I do know the difference between "Square" and "squared". But that only works for squares - for any other shape, "unit squared" is wrong, it should be "square units". Similarly, your suggestion that the area of an 8x8 room is "64 feet squared" is also wrong. You could say "8 feet squared" (ugh) or "8 feet square" (not often used) - but both only work for squares.
I'm a math illiterate. So here's my stupid question. Why can't the base be 9 and the height be 8? 9^2+8^2 (or 81+64) = 145 and the hypotenuse squared is 144. 1/2(9)(8) = 36. Why is this wrong?
No, you can’t be categorized as a math illiterate or you are asking a stupid question! It is an interesting question and it is always great to ask questions. How many people don’t ask questions to protect their ego? Actually, I think you are helping so many people by asking a question for everyone to learn 🙏🏽 I am not an expert, but my understanding is as follows: I think 145 is different from 144 and the difference is not negligible. Also, 8 is different from 11.4708 and 9 is different from 3.5084. So, the areas calculated in the two different scenarios are different, such as 36 and 20.
The minor error, as you've already noticed, is that the hypotenuse of your triangle would be very slightly more than 12, because 8² + 9² is very slightly more than 12². The much more significant error is that in your triangle the angle between the hypotenuse and the base would not be anything like 17 degrees. It would be a little over 41.6 degrees. (Edited to add: it's not a stupid question. It's a perfectly reasonable question).
The simplest, most straightforward, means of solving is to multiply the hypotenuse (12) by sin(17), multiply the hypotenuse by cos(17), then multiply the aforementioned results together, and finally divide by 2. The result is approximately 20, which is what many of the other comments herein had as an answer.
I agree. You and I are of one mind.
I would like to practice with you.
Right, you are using the area of a triangle formula 1/2 ab sin(c).
Therefore: area=1/2 x 12 x 12 x cos (17) x sin (17) = 20.13 square units.
@@adamclark1972uk1 mind for 2 heads ? 😂
As long as we are using trig anyway, just get the length of the other side as 12cos(17), easier to me
Area = (1/2)(base)(height) = (1/2)(12cos[17])(12sin[17])
= (1/2)(144)(cos[17])(sin[17])
= (1/2)(72)(2)(cos[17])(sin[17]), but sin(2x) = 2sin(x)cos(x)
= (1/2)(72)sin(34) = 20.13094453... (approx 20.131 units^2)
Like others, I'm surprised you went with Pythagoras for this. If you can do 12 sin 17° then you can do 12 cos 17° just as easily, and now you have base and height. So the area is:
½ × (12 sin 17°) × (12 cos 17°)
=
72(sin 17°)(cos 17°)
From the thumbnail, I thought the point of this video was going to be too teach the trig identity (which i had to look up just now) that (sin X)(cos X) = ½ sin (2X), so here we're just doing 36 × sin 34°.
That seems a much cleaner way to get to the answer.
Due to your round off error, in an application of trig in a real world machine shop, your answer is off by 0.004 thousandths and by 0.0039 ten thousandths depending on the called out +/- tolerance, which is about the thinkness of a piece of copy paper or average human hair. It all depends on round off error
Why couldn’t I have had a math teacher like you when I was in School to have the patience as you do and be able to dumb it down to as to absorb so easily. Thanks
Like others have said why not use the cosine to find the other side
Unless the point was to teach the pythagorean theory.
I think it's better to stick to one topic
Those who know how to use sine, cosine and tangent in a right triangle already know the Pythagorean theorem: that is basis geometry.
This is easy. Everyone knows that sin theta = opposite side/hypoteneuse. Since theta = 17 dgrees, sin17 = opposite/ 12. Opposite side = sin 17 degrees * 12 = 3.508; cos theta = adjacent / hypotoneuse. cos17 = adjacent/12. adjacent side = cos17 *12 =5.8. So, area of triangle = .5*b*h. So, area of this triangle = .5 *cos17*sin17*(12)^2 =20.17 units^2
I’m a penn state fan but I have no idea what you are talking 4:00 About. I got a phd in English and I build violins using simple geometry. Everyone doesn’t know this formula. Ha haha. But I hope The football coaches can compute.
WOW...got it 20.1 SOHCAHTOA and Pythag thanks for the fun
A trig question:
Sin 17° = a/12
a = 12(Sin17)
= 12(0.2924)
= 3.5085
b =?
tan17° = a/b
b = a/tan17
= 3.5085/(0 3057)
= 11.4758
Area triangle:
A = (1/2)b×h
Here
b = b = 11.4758
h = a = 3.5085
A=(1/2)(11.4758)(3.5085)
=(1/2)(40.2628)
=20.1314 units^2
Verify...
11.4758^2+3.5084^2=?17^2
131.6940+12.3089=?289
144.0029=❌️289❌️
??
= 12^2. Not. =17^2
Where does the 17 come from? The hypotenuse length is 12, no?
@@francisdelpuech6415 my bad
17 is 17 degrees, the measure of the angle.
The hypotenuse is 12.
So, applying a^2 + b^2 = c^2,
11.4708^2 + 3.5084^2 = 12^2 = 144
The way I did ot is system of equation which is 12=square root of x²+y² then inverse tan(x/y)=17 this is a picture of a 12 radius circle with a line through it in which where the line touchs the edge of the circle the missing sides are the x and y that make the line intersection. This came to me naturally in thought, a interesting way to find the sides of a right angle with hypotnuse and one angle using algebra. In which you do a simpe area formula 1/2×11.47×3.508=20.1309.....
Your videos are a godsend.
Thank you so much 👍
First thing first, from the image, we can elicit all the angles measuse as it is a right triangle so there must be one 90 degree angle which is marked in the right bottom, then addressed 17degree angle.
From prior knowledge we know, triangle consist of 180 degree, so the top angle should be 180-(90+17)=73 degrees.
Now, apply soh-cah-toa over this right triangle to find out the side lengths first then, we will conclude it to the area of this rectangle.
apply soh at 17 degree angle,
=> sin 17 = x(which is vertical side length or height)/12(hypotenuse)
=>12 sin 17 = x
=> x = 3.508 which is similar to 3.51
its time to find out the base value as we don't know,
apply 'toa' criterion,
tan 73 = y(base length)/3.51
=>y = 3.51 tan 73
=> y = 11.48
As we find out base and height of this triangle, so now we can place it in the triangle area formula,
(1/2)*x*y
=0.5*3.51*11.48
=20.15
If you are going to use the sine of 17 degrees to find the opposite side, why not just use the cosine to find the adjacent side and you don't have to do all the squaring and rooting?
thank you, teach. good review. Thanks again.
72 sin(17) cos (17)
I cheated. I did a quick Pythagorean based upon 144 no calculator. 11.5 X 3.5 worked,, so my guesstimate was 20 square inches area. I LIKED your explanation of sin. A LOT.
Was it supposed to be 11.4708^2 + 3.5084^2. = 12^2 = 144 ?
It should be a^2 + b^2. = c^2
@@chrisdissanayake6979 Something like that.. You have the idea. Like I said,, I cheated. Some much too quick mental math and the first answer I came up with was actually 40.17,, but then I remembered divide by 2 Carpenter. For new house layouts, squaring with corner to corner measurements,, or to check a bath for squareness before cutting tile,, almost second nature for me. I DO almost always pull out a calculator to double check my mental math before I start cutting and nailing,, cowardice. I remember the carpenter phase, "I've cut it three times and it is still too short." Also,, when doing boringly repetitive work,, rolling out sod on new construction, or nailing shingles on a roof.. ??? You continually chant Green side up. Green side up,, Gree,,, Confession,, I have laid a roll of sod brown side up one time,, hot,, tired to stupidity,, Did not notice till I came back with the next roll and had to think about why it looked odd.
(12*cos(17))×(12×sin(17)/2=20.130 is the answer
Approximately 20.130944524946885885775413039496
Area = 20
Okay Mr John,
20 units^2
It is incorrect to say "20 units squared" vs "20 square units" - the two are not the same. If you have a square that is 8 inches on a side, what is its area? It is "64 square inches", not "64 inches squared". Semantics matter, sometimes.
@@greenmanofkent
I thought feet^2 is another way of writing square feet.
@@chrisdissanayake6979as are in², cm², mi².
Where we are getting confused is this: consider a room which is a square with all sides 8 feet long. We can state the room size as 8'×8', or we might say 8 feet square. Saying 8 feet squared is not technically correct, but is all too commonly used, even though feet squared is a measure of area.
@@richardhole8429
thank you 🙏🏽
@@chrisdissanayake6979 you are correct. The post you replied to is in error. He is confusing "unit square" with "unit squared". A room 8 feet square is 8x8. A room of 64 square feet, or 64 feet squared, could also be 8x8, or 16x4, etc. Because unit squared and unit square are pronounced similarly, it is preferable to say square units.
@@richardhole8429 I am not in error. I do know the difference between "Square" and "squared". But that only works for squares - for any other shape, "unit squared" is wrong, it should be "square units". Similarly, your suggestion that the area of an 8x8 room is "64 feet squared" is also wrong. You could say "8 feet squared" (ugh) or "8 feet square" (not often used) - but both only work for squares.
I come up with approximately 20.13094453 square cubits. (Because he didn't say inches or feet.)
20-1/8
2.127 square
I got 20.08 does it still count ?!!
20.13 sq ft
Lordy,this is why I chose Political Science instead...at least I could talk my way out of problems.
Jesus Christ saves today! Turn to God. Repent. The day of judgment is near!
You bore me to death, honestly
I'm a math illiterate. So here's my stupid question. Why can't the base be 9 and the height be 8? 9^2+8^2 (or 81+64) = 145 and the hypotenuse squared is 144. 1/2(9)(8) = 36. Why is this wrong?
No, you can’t be categorized as a math illiterate or you are asking a stupid question!
It is an interesting question and it is always great to ask questions.
How many people don’t ask questions to protect their ego?
Actually, I think you are helping so many people by asking a question for everyone to learn 🙏🏽
I am not an expert, but my understanding is as follows:
I think 145 is different from 144 and the difference is not negligible.
Also, 8 is different from 11.4708 and 9 is different from 3.5084.
So, the areas calculated in the two
different scenarios
are different, such as 36 and 20.
The angle of 17 degrees dictates what the sides of the triangle are. You can't just arbitrarily give the sides any length.
@@lwh7301 Thank you 🙏🏽
That really makes sense!
The minor error, as you've already noticed, is that the hypotenuse of your triangle would be very slightly more than 12, because 8² + 9² is very slightly more than 12².
The much more significant error is that in your triangle the angle between the hypotenuse and the base would not be anything like 17 degrees. It would be a little over 41.6 degrees.
(Edited to add: it's not a stupid question. It's a perfectly reasonable question).
@@gavindeane3670
Thank you so much!