Estimating arctan, arcsin, arccos

[zell_D]

I have basically arrived at the answer of arctan(3) = theta, but the thing is, we are not allowed to use a calculator and the choices are numbers with decimals. I have no idea how to estimate these things, is this even possible?

 

[harimakenji]

I am not sure how to solve this without calculator...maybe using trigonometry table?

 

[zell_D]

well this is on a KaplanDAT practice problem and i can't get it. and they don't explain it either so i don't know lol

 

[DorianG]

Are the possible answers very far apart so there is only one logical answer?

I don't know how this would be solved without a calculator or log tables unless this is the case.

Oh, also if you use a calculator, is the correct answer on the list of possible answers? Just to check if you've arrived at the right expression with arctan 3 = theta

 

[zell_D]

the possible answers are very apart other than two of the answers. one of which contains the right answer, one around 71.6 (the right answer) and hte other 79.3 or close to those numbers.

 

[tiny-tim]

Hi zell_D!

I have basically arrived at the answer of arctan(3) = theta, but the thing is, we are not allowed to use a calculator and the choices are numbers with decimals.

erm … just draw the triangle!

(and estimate the angle from that )

 

[DorianG]

Realised this about an hour after posting this morning!

As tiny-tim says, draw the triangle remembering that tan is opposite over adjacent. Use Pythagoras and basic trig to calculate the angle!

 

[tiny-tim]

Use Pythagoras and basic trig to calculate the angle!

uhh?

just measure it! ​

(alternatively, use sinx < x < tanx for small x )

 

[zell_D]

how would i measure it though without a measuring tool? and estimating/eyeballing it wouldn't give me an answer to separate the two answers within the 70's.

 

[Phrak]

What is you set of possible answers? And are you allowed to use a straight edge and compass?

 

[Feldoh]

Well depending if you in degrees or radians arctan(x) ranges from -90 degrees to + 90 degrees (-pi/2 to pi/2) so if there are any answers out of that range you can eliminate them immediately.

arctan(1) = 45 degrees or pi/4

So you know that You answer must be between 45-90 degrees or pi/4 to pi/2

Since the first derivative is locally spiked near x=0 and arctan(0) = 0 I suppose you could approximate the integral of 1/(x^2+1) on [0,3]. If you take the average of the left and right-end point approximations witha partition of 1 you get an answer of 1.25 and the real answer is like 1.249045 so it's relatively close -- this is probably how I would do it if a lot of the answers were on the interval pi/4 to pi/2.

But you're trying to do this really fast right? I don't think there's any really good way to evaluate that. Can't to a taylor series and take the first few terms because it converges on (-1,1)

 

[DorianG]

Can't you use basic trig to find that the length of the hypotenuse of a 3 - 1 sided right triangle is 1² + 3² = 10, so the hypotenuse is 10^1/2.
Then you can use the cosine or sine rule? Maybe the cosine is best, with three sides known and one angle wanted?

 

[tiny-tim]

how would i measure it though without a measuring tool? and estimating/eyeballing it wouldn't give me an answer to separate the two answers within the 70's.

How about the complementary angle, which will be in the 10's?

(or sinx < x < tanx for that angle)