# Trigonometry

• I know the formulas for finding sine, cosine, and tangent, but how does a calculator find these functions without some of the parts in the formula? This means that say if I enter sine 50, the calculator gives out a response, but I did not give it the opposite or the hypotenuse's length. How does the calculator still manage to give a response?

• @the-blade-dancer I don't really know, but I think it takes a right triangle with the number you entered as an angle.

Like if you put in 50, it'll take the sine/cosine/tangent of the angle that's 50 in a triangle with angles 50, 90 and 40.

• How does it calculate the lengths tho? It feels like angles can do something, but idk what. Also what about when you enter numbers larger than 180, like 700? Apparently that still works

• @the-blade-dancer I have no idea how larger angles work. Also I'm not sure but I think intuitively if two triangles have the same angles the ratios of their sides are the same.

(I have no idea what I'm talking about here, this is just a guess.)

• @The-Blade-Dancer Every function, like sine or cosine, can be approximated with a polynomial, using calculus or something, I don't know. The calculator then just plugs in the number into the approximate polynomial and outputs the answer.

Another way is to roughen the edges; thinking the sine/cosine wave as a bunch of little lines connected to each other.

Also if you enter larger angles you'll have to resort to the unit circle definition.

• In the unit circle, things repeat every 360(2pi radians) degrees. So you technically find how many 360's are inside of that large angle and find the remainder when divided.

If you are using angles greater than 90 degrees, then you are not in the first quadrant anymore. Using the unit circle definition, the cosine of the angle is the x-coordinate when it is on the unit circle and the sine of the angle is the y-coordinate. So it is possible to have negative sines and cosines. The tangent is simply the slope of the line.

I hope this explanation helps • Interesting