Solving trigonometry problems

How to do trigonometry problems depends on the type of problem that we are facing.

Solving for parts of right triangle:
First let us see how to solve trigonometry problems that involve parts of a right triangle. In a right triangle, we have one right angle and two acute angles. Suppose if we name one of the acute angles as “theta”, then the side opposite of theta is named the ‘opposite side’. The sides that form the angle theta would be the hypotenuse and third side would be called ‘adjacent side’. These are now used to define the three main trigonometric ratios: Sine of angle theta, Cosine of angle theta and Tangent of angle theta. These ratios are defined as follows:

Sine (theta) = Sin(theta) = (opposite side)/(hypotenuse)
Cosine(theta) = Cos(theta) = (adjacent side)/(hypotenuse)
Tangent(theta) = Tan(theta) = (opposite side)/(adjacent side)

Therefore if we know any two sides and one of the angles, then we can find the remaining sides and angles.

Solving for parts of triangles that are not right triangles:
If you need help with trigonometry problems, when the triangle is not a right triangle, then you would need to use the sine or the cosine rule. For being able to use the cosine rule you would need to know any two sides of a triangle and the included angle. To be able to use the sine rule, one would need to know at least one pair of side and its opposite angle and any other side or angle, then the remaining three parts of the triangle can be calculated. So basically we need to know at least three parts of a triangle and then the remaining three parts can be calculated using the sine or cosine rules.

Solving trigonometric identities:
The primary trigonometric identities, the secondary trigonometric identities and the inter relationships between trigonometric ratios are used to solve trigonometric identities. The best way to approach a trigonometric identity proving problem is to convert the left side to sine and cosine functions. Then simplify using basic algebra and then finally use the other trigonometric identities and other trigonometric ratios to simplify further.

Finding angles using trigonometric ratios:
These types of problems are different from the ones we discussed before. These involve working backwards from the trigonometric ratio to get the angle. We call them inverse trigonometric functions.