I am wondering if there is a way to figure out the interior angles of a right triangle if we know ONLY the side lengths, and the trick is, we CANNOT use arctangent!

Greg

Hi there,

Arctangent is only one of the three methods involving trigonometric ratios, so if it is off-limits, use arcsine or arccossine instead.

The sine ratio is opposite/hypotenuse and the cosine ratio is adjacent/hypotenuse.

Using one of these will result in the measure of one of the acute angles in the triangle. To find the third, use the property that says the interior angles of any triangle add up to 180 degrees. Since one is 90 degrees in a right triangle, and you have just determined the measure of one of the acute angles, the third angle is easy to find!

Hope this helps,
Leeanne

Greg wrote back,

I wish your answer was that simple, but there are no inverse tools I can use. Is there another way?!

Greg

Greg,

You could approximate the value, for example using Newton's Method.

Newton's Method is a technique to approximate a solution to the equation f(x) = 0. It produces a sequence of approximations x0, x1, x2,... where x0 is an initial approximation, and after that the values are given by

xn+1 = xn - f(xn)/f'(xn)

where f'(x) is the derivative of f(x).

The hope is that the sequence of values x0, x1, x2,... approaches a solution to f(x) = 0, but Newton's Method doesn't always behave well, it depends on the form of the function f(x) and the initial value chosen x0. Happily the tan(x) for x between 0 and 90 degrees is very well behaved and Newton's Method will work for you. Since the expression contains the derivative of the tangent function it is more natural to work in radians rather than degrees.

Suppose that the side lengths of the triangle are a and b and you want to find the angle x (in radians) where tan(x) = a/b. Let

f(x) = tan(x) - a/b

then

f'(x) = sec2(x) = 1/cos2(x)

Newton's expression then becomes

xn+1 = xn - cos2(xn) ( tan(xn) - a/b)

Any reasonable initial value will work, in the example below I used 45o ( /4 radians).

For an example I took a = 1 and b = 3 so I am looking for x such that tan(x) = 1/3 and I took x0 = /4 which my calculator gave me as 0.785398 radians. Here are my calculations.

 xn xn+1 f(xn+1) n = 0 0.785398 0.452065 0.152271 n = 1 0.452064 0.328849 0.007906 n = 2 0.328849 0.321768 0.000019 n = 3 0.321768 0.321751 0.000000

When you decide to stop is your choice, it depends on the accuracy you want. Once you have a solution x in radians you can convert it to degrees by

x radians = x 180/ degrees

so in my example

0.321751 radians = 0.321751 180/ = 18.4350 degrees

I hope this helps,
Penny