# find the sine cosine and tangent

• Jan 23rd 2010, 07:10 AM
GoldenBoy
find the sine cosine and tangent
My question as follows derives from:

sin(x)=a/c;

given values a=3 and c=5 just for examples sake:

sin(x)=3/5=0.6.

ok, so my question is, knowing the sine of the angle i would like to calculate the angle in degrees and in radians.

The trick is I don't want to use a calculator.

I'm guessing there is an algorithm to calculate the angle. What is it and how would it be applied to my example?
• Jan 23rd 2010, 09:33 AM
Hello GoldenBoy

Welcome to Math Help Forum!
Quote:

Originally Posted by GoldenBoy
My question as follows derives from:

sin(x)=a/c;

given values a=3 and c=5 just for examples sake:

sin(x)=3/5=0.6.

ok, so my question is, knowing the sine of the angle i would like to calculate the angle in degrees and in radians.

The trick is I don't want to use a calculator.

I'm guessing there is an algorithm to calculate the angle. What is it and how would it be applied to my example?

You'll have to use a series expansion of arcsin which you'll find here.

This will give the angle in radians, if you know the sine of the angle.

If you look at the attachment, you'll see I've used this series in a spreadsheet to demonstrate how the answer builds up as you add each term.

In column B I've worked out each individual term, and in column C the accumulated totals. You'll see that the total, for z = 0.6, is approaching 0.643501, which is the answer correct to 6 d.p. The more terms you take, the more accurate the answer. But it will be hard work without a calculator!

• Jan 23rd 2010, 11:42 AM
GoldenBoy

Ive looked over the ecuations you linked and my results coincided with yours.

Thankyou.
• Jan 25th 2010, 07:35 AM
GoldenBoy
well i used the arcsin function and it works relatively well up to arcsin(0.8) in not too many steps. the problem ocurrs at arcsin(0.9) which loses 1º in the conversion and at arcsin(1) (which should be 90º) gives 77.7º or 1.35rads and i cant get passed it.

I even passed the formula to a scientific calculator which accepts equation but no luck.

what am i doing wrong?
• Jan 25th 2010, 09:12 AM
Hello GoldenBoy
Quote:

Originally Posted by GoldenBoy
well i used the arcsin function and it works relatively well up to arcsin(0.8) in not too many steps. the problem ocurrs at arcsin(0.9) which loses 1º in the conversion and at arcsin(1) (which should be 90º) gives 77.7º or 1.35rads and i cant get passed it.

I even passed the formula to a scientific calculator which accepts equation but no luck.

what am i doing wrong?

Yes, the series converges very slowly as $z \to 1$. So you're best to use the alternative series on that Wikipedia page for $\arctan(z)$:
$\arctan(z)=z-\frac{z^3}{3}+\frac{z^5}{5}-...$
and use the formula (above that one on the same page):
$\arcsin x = 2\arctan\left(\frac{x}{1+\sqrt{1-x^2}}\right)$
to convert one angle to the other.

I attach a spreadsheet extract showing the comparison between the two methods. Method 1 uses the angle $0.9$ radians in cell B1, and Method 2 uses the corresponding angle (using the second formula above) $0.626789$ in cell D1. You'll see that this method converges much more rapidly to the 'correct' answer of $1.11977...$