Given one trig value for an angle, how to find the other two.

 

We have learned The Trigonometric Identity, 

sin² theta + cos² theta = 1. 

It's a simple application of the Pythagorean Theorem. a² + b² = c² where c = 1. This means if either the sine or cosine of an angle is known, we can find the other value easily enough. Simply square the value you know and subtract that value from 1, then take the square root of the value. This is where our practice with square roots and fractions will come in handy.

Example: sin theta = 1/5

(1/5)² + cos² theta = 1

cos² theta = 1 - 1/25 = 24/25

Taking square roots, we get

cos theta = sqrt(24/25) = sqrt(24)/5 = 2sqrt(6)/5

And, of course, if we know sine and cosine, tangent = sine/cosine.

sin theta = 1/5, cos theta = 2sqrt(6)/5, so tangent = 1/2sqrt(6) or sqrt(6)/12.

sub Example: cos theta = 1/5

It's the exact same work, just finding sine instead of cosine.

sin² theta + (1/5)²   = 1

sin² theta = 1 - 1/25 = 24/25

Taking square roots, we get

sin theta = sqrt(24/25) = sqrt(24)/5 = 2sqrt(6)/5

cos theta = 1/5, sin theta = 2sqrt(6)/5, so tangent = 2sqrt(6).

  

Finding the sine and cosine when we know tangent is different and a little trickier since sin theta/cos theta = tan theta, multiplying by cosine on both sides gives us

sin theta = tan thetacos theta.

We make a substitution into The Trigonometric Identity, do some algebraic manipulation and we can find cosine.  Once we have cosine and tangent, we multiply them together to get sine.

Example: tan theta = 2.

This says that 2²cos² theta + cos² theta = 1

5cos² theta = 1

cos² theta = 1/5

cos theta = sqrt(1/5) = sqrt(5)/5.

multiply cos and tan and we get sin theta = 2sqrt(5)/5.


Here are some practice problems.  Answers are in the comments.


a) sin theta = 1/3

b) cos theta = 1/3

c) tan theta = 1/3