COMS Tutorial Corner

Isn’t that cool? Finding Angle(s) on a Calculator Without Trig Functions

Do you know there is a simple empirical formula finding the angles, giving three sides a, b, c of any right angle? Let a < b < c where c is the hypotenuse. The angle A (which shall be the smaller acute angle) shall be:

A is approximately (86 a) ⁄ {(b/2) + c} (note the result is in degree) — (**)

Let’s look at a few examples to verify that the formula above does give us the correct result.

For 1-1-√̅2 triangle, both of the acute angles shall be 45 degrees. While using formula (**), we get:
A is approximately (in degrees):: (86 x 1) ⁄ {(1/2) + √̅2} = 44.9
For 1- √̅3 -2 triangle, the smaller acute angle is 30 degree. While using formula (**), we get:
A is approximately (in degrees):: (86 x 1) ⁄ {√̅3/2 + 2} = 30.01
— So that’s pretty close.

Finally, let us look at the famous 3-4-5 right triangle: with the given formula,
A is approximately (in degrees) (86 x 3) ⁄ {(4/2) + 5} = 36.86
You can verify the smaller acute angle of such a triangle is roughly 36.87 degree.

Wondering how do we get the constant 86? You can find this constant from the formula:
(3/2) x (180 ⁄ π) = 86

An alternative formula that gives you roughly the same result is:

A is approximately (3 a) ⁄ {b + 2 c} x (180 degree /π) — (*)

Question for thought:

1) With this formula, can we also find the larger acute angle of the right triangle?

2) We applied the formula to find the smaller acute angle of a right triangle. Can we apply this formula to any triangle?

3) Do the calculation yourself – find the smaller acute angle respectively for the following two right triangles:
3a) the 1-2-√̅5 triangle
3b) the 8-15-17 triangle

Wow! With the approximate formula as given in this article, you can find the angle, on a calculator even without a trigonometry / inverse trigonometry functionality. Isn’t that cool?

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