Given the following figure, what is theta? I feel like I should be able to get this but my trig skills are failing me. Please show your work, so I can understand the solution.
A trig problem for your entertainment
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I mean, you're dealing with right angle triangles... Since the total of all angles added together is 180 degrees, you've got a known angle (90), now you just need to pick some lengths and using the math you should be able to compute the other two angles.
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Correct, these are alternate interior angles and the adjacent angles are equivalent via congruence.
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You used "Trig" and "Entertainment" in a sentence without the important linking phrase "is not".
At times like these I draw it in Inkscape and use the 'ruler' tool which gives me the angle.
At times like these I draw it in Inkscape and use the 'ruler' tool which gives me the angle.
KevinT
Well-Known Member
Does this work? Looking at the drawing sideways.
At theta = 0 the delta y is 1.283 - 0.991 = 0.292,
Then at theta = ?, x=3.0 and y=.292
Resulting in arcTAN(theta) = .292 / 3.0
Theta = 5.56 deg
At theta = 0 the delta y is 1.283 - 0.991 = 0.292,
Then at theta = ?, x=3.0 and y=.292
Resulting in arcTAN(theta) = .292 / 3.0
Theta = 5.56 deg
NateB
Well-Known Member
I was looking for a small protractor in my desk at work, but we don't have one out here.
I use the ruler tool a lot for even simple measurements. It takes away at least one potential problem (me) when working with numbers...
Hyp of rect (h) = sqrt(3^2 + 1.283^2) = 3.262834504
If you draw a circle from the bottom right corner of the rectangle with radius (r) 0.991 and then take a tangent line from that circle to the top left corner - you don't need to know the length of this line as we already have length of the hyp of the rectangle and radius of the circle. So, we can gather the angle of that tangent line and the hyp: arcsin(r/h) = 0.308598507 radians or 17.68139202 deg
We can gather the hyp angle of the rectangle by arccos(long side/h) = arccos(3/3.262834504) = 0.404127168 radians or 23.15478112 deg
Now the easy part - subtract the previous angle from the hyp angle = 0.404127168 - 0.308598507 = 0.095528661 radians or 5.473389103 deg
Brian S was correct.
TP
If you draw a circle from the bottom right corner of the rectangle with radius (r) 0.991 and then take a tangent line from that circle to the top left corner - you don't need to know the length of this line as we already have length of the hyp of the rectangle and radius of the circle. So, we can gather the angle of that tangent line and the hyp: arcsin(r/h) = 0.308598507 radians or 17.68139202 deg
We can gather the hyp angle of the rectangle by arccos(long side/h) = arccos(3/3.262834504) = 0.404127168 radians or 23.15478112 deg
Now the easy part - subtract the previous angle from the hyp angle = 0.404127168 - 0.308598507 = 0.095528661 radians or 5.473389103 deg
Brian S was correct.
TP
Thank you for all the answers so far. I should have clarified up front that I am looking for an analytical solution. Thank you K'Tesh for point out that showing theta in the other locations would have made for a much simpler and clearer drawing, no idea why I didn't do that in the first place.
Given the small angle, it works well as a very easy-to-calculate approximation, and just saying "around 5.5 degrees" would be more than enough accuracy for my needs, but it is not a precise solution.
Did you solve this analytically or numerically or empirically? If analytically, can you describe how you did it?
I apologize but I'm having trouble understanding exactly where you're drawing the circle and the tangent, and without that I can't follow the rest. If you could provide a simple picture it would be a big help.
hypotenuse - pink
hypotenuse angle - brown
Tangent Line (dotted blue line from circle tangent to top left corner of box)
tangent line <> hypotenuse angle - light grey
Circle Radius - cyan
theta (answer) - white
Hope that helps,
TP
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Maybe it was his wording. He said to draw a circle that was R0.991" from the lower right corner and the left red line (he posted a drawing right before me. Red in your sketch, blue dotted in his sketch) was tangent to that (i.e. its a right angle, so right triangle math works). You could skip drawing the circle and just use the exact same arcsin he said if you saw the triangle. I do draw circles myself, as sometimes seeing the tangent reminds me that you can use either rectangular coordinates (Pythagorean theorem) or polar coordinates. Its all the same, but sometimes seeing the circle can let you skip a step or two (or at least keep you on track without jumbling math).
Sandy.
Thanks. On further question, though... in your picture the angle t is shown between the hypotenuse and the blue dotted line. On the one hand, you say the blue line is the tangent line that intersects in the upper left corner... but as drawn it does not. It is one of the sides of the canted rectangle, whose upper left corner is not the same as the the one where the hypotenuse terminates. The blue dotted line is correct as drawn, because that properly defines theta.
And so therefore the angle of the theta + t is not quite equal to the angle of the hypotenuse (it's very close, but not exact.)
Apologies if I am misunderstanding what you wrote (or drew), but I can't figure out any other way to interpret it.
Let me change that dark blue dotted line to a nice bright solid yellow one:
Zoomed in:
note: if that line wasn't extending to the upper corner of the rectangle, then I couldn't really utilise the hypotenuse of the rectangle.
Of course, as it turns out, it passes through the upper left corners of both rectangles.
TP
Zoomed in:
note: if that line wasn't extending to the upper corner of the rectangle, then I couldn't really utilise the hypotenuse of the rectangle.
Of course, as it turns out, it passes through the upper left corners of both rectangles.
TP
Is your diagram suppose to illustrate it passing through it? If it doesn't pass through it, we really need the distance it misses by.
Scrub that, it must be passing through it. Otherwise the 0.991 dimension would be meaningless. Just one of those inexact raster type drawings where lines are drawn close enough for purpose, but not exact.
TP
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Using K'Tesh's drawing highlighting the alternate angles = Theta. Specifically the one on far right.
Side opposite Theta = (1.283-0.991)" = 0.292" = opp
Side adjacent Theta = 3" = adj
Tan Theta = opp/adj
Theta = arctan(opposite/adj) = arctan(0.292/3) = arctan(0.09733333333)
Theta = 5.55927755238 degrees approximately only due to the repeating "3" beginning in the 1/10,000ths place. The answer has three reliable digits to right of decimal, so rounding from there to hundredths seems accurate (enough) to me.
Thus, Theta is approximately 5.56 degrees.
Pardon the formatting.... I did this in bed, in the dark, at 0500hrs on my phone.
Oops, I did not read far enough.... KevinT used the same geometry/trig. that I did, and of course gave you the answer first. Well done Kevin.
Thank you K'Tesh for the use of your concise diagram.
A long, LONG time since I did that problem.... 1981. Thank you W.J. Harlan for being such an outstanding teacher.
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I had to go back and look at our respective drawings more carefully to figure out the discrepancy. One aspect of your drawing threw me off, but now I realize it was a result of my initial bad choice on where to show theta (rather than the more natural location that K'Tesh showed).
So as far as I'm concerned your solution is correct.
Thanks for talking me through it. Thanks also to @StreuB1 for the correct answer.Now I have to look at @cautery and @KevinT 's answers in more detail and see why they're getting a different result.
I had to go back and look at our respective drawings more carefully to figure out the discrepancy. One aspect of your drawing threw me off, but now I realize it was a result of my initial bad choice on where to show theta (rather than the more natural location that K'Tesh showed).
So as far as I'm concerned your solution is correct.
Now I have to look at @cautery and @KevinT 's answers in more detail and see why they're getting a different result.
Maybe I can clarify our reasoning without mucking up the water too much.
First, I am operating on a number of assumptions that were not clarified in the original problem....
1) lines bounding 1.283" and 0.991" dimensions respectively are parallel to each other.
2) left boundary lines on both dimensions intersect the same point at the top there....
3) other...?
Thus, the side opposite Theta becomes a simple line segment difference problem....
The side adjacent to Theta is explicitly called out as 3".
By definition, the tangent of any acute angle in a right triangle is equal to the length of the opposite side divided by the length of the adjacent side.
This is a simple ratio problem, but results in an approximate result due to the repeating sequence beginning in the 4th place to the right of the decimal.
But taking the arctan on the extended approximation preserves the 3 digit significance limit created by the 0.292 numerator in the ratio.
The PROPER logic to use in rounding the final answer approximation is to round down to three significant digits.... My logic while giving the correct approximation was more intuition than logoc.
Sorry, I 'be been out of the everyday trig game for a long time.
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Trig is only entertainment when it happens to someone else.
lakeroadster
When in doubt... build hell-for-stout!
Assumptions are correct. Here, however, appears to be the error:
The black segment at the bottom left or top right has
Actually: 1.283 - (.991*cos(theta)), slightly larger than 0.292.
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I am out of town, so I don't have the ability to draw my "proof", and I am not inclined to argue over 9/100ths of 1 degree.
Sufficient to say that I disagree...
Sure was fun exercising the brain....
Curious, source and purpose of the problem?
Sorry, I got that wrong. The length of the referenced black segment is actually 1.283 - (.991*cos(theta)), which is slightly *larger* than 0.292, which is why your answer is coming out slightly higher than correct.
A potential upcoming build.
You have most definitely piqued my interest.
And now I have to go figure out if and how I went wrong. It will drive me to distraction until I do.
Just focus on that one value, how you got 0.292, despite the angle.And now I have to go figure out if and how I went wrong. It will drive me to distraction until I do.
EDIT: LOL I got it wrong again. Third try:
Using the known correct value of theta, the short segment works out to 0.287, although you could not calculate this until you had correctly calculated theta.
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Yep, I see it now. I was just about to draw it out with extraneous info excluded.
Absolutely obvious and and a little embarrassing that I let it get by me. I am clearly rusty.
Thank you for your patience....
lakeroadster
When in doubt... build hell-for-stout!
Which is probably close enough... well... unless it's Rocket Science.
