Question

How do you use trigonometry rations to solve for a missing side or angle of a right triangle?

Answer 1

In your questions you are always given at least one known side and one unknown side with a known angle, or an unknown angle and two known sides.

To solve for the unknown side, mark this side and the known side as either hypotenuse, adjacent or opposite side with respect to the known angle. Then find out which trig ratio sin, cos or tan involves those two sides. Use SOHCAHTOA to help.

Then write an equation involving the selected ratio and the sides and angle.

For example:



You are required to find x. To do so, we must label x and the other known side as H, O or A.

Answer 2

We see we have an opposite and adjacent for angle a, thus we must use tan since only tangent ratio involves O and A. The others require the hypotenuse which is not given.

Recall that tan a = O/A
So tan a = x/20 which rearranges to x = 20 tan a. Then just sub in whatever angle 'a' is.

To find a missing angle, it's actually a bit easier. You again must label the two given sides as H, O or A and then pick the right ratio depending on what pair of sides you have.
Say you have O = 15m and H = 30m.
This involves sin a = O/H = 15/30 = 1/2
This tells us the sine of the missing angle gives 1/2.
To find the angle now, we use the inverse operation of sine, called inverse sine. Taking inverse sine of the RHS, 1/2 gives the angle required.

\[a = \sin^{-1} 1/2 = 30^{o}\]