In a right triangle, we haev some trigonometric relationships between the sides and angles. Given an angle, the ratio between the opposite side to the angle by the hypotenuse is the sine of this angle, therefore, the following statement
Describes the following triangle
To find the missing length x, we could use the Pythagorean Theorem. The sum of the squares of the legs is equal to the square of the hypotenuse. From this, we have the following equation
Solving for x, we have
The missing length of the first triangle is equal to 4.
For the other triangle, instead of a sine we have a tangent relation. Given an angle in a right triangle, its tanget is equal to the ratio between the opposite side and adjacent side.The following expression
Describes the following triangle
Using the Pythagorean Theorem again, we have
Solving for h, we have
The missing side measure is equal to 13.
Now that we have all sides of both triangles, we can construct any trigonometric relation for those angles.
The sine is the ratio between the opposite side and the hypotenuse, and the cosine is the ratio between the adjacent side and the hypotenuse, therefore, we have the following relations for our angles
To calculate the sine and cosine of the sum
We can use the following identities
Using those identities in our problem, we're going to have
