4.3 Right Triangle Trigonometry

Trigonometric Functions

In any right triangle, all three sides have a relation with any of the 3 angles. Relative to angle x, these three angles are either adjacent to angle x, opposite of angle x, or the hypotenuse of the triangle.

Using this principle, 6 different ratios can be created. It is also important to note that angle x is greater than 0 degrees and less than 90 degrees.

SOH-CAH-TOA

                      

                      

The functions cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively. These are known as Reciprocal Identities.

                             

The cotangent function and tangent are known as Quotient Identities

       

This information, along with the Pythagorean Theorem can be used to to find the Pythagorean Identities:

          

Lets derive the first identity:  by using the Pythagorean theorem and our knowledge of the trigonometric functions. 

           

           

Trigonometry and the Unit Circle

Looking at the first quadrant of the unit circle, imagine that a right triangle is created with an angle measure of  at the origin. Using the property triangles, it can be inferred that the remaining angle is . By using the trigonometric functions, either leg of the triangle can be determined. To find the leg opposite of angle x, you can use the sine function to determine the side length. 

     Because we know that angle x is , we replace x with the degree of the angle. Since the the radius of the unit is 1 unit, we can say that the hypotenuse of the triangle is 1. 

      Here we can see that the sine of  is equal to the side opposite of the angle. This  value also corresponds to the y value of the point that lies on the unit circle. Therefore,  is equal to the y value of the point created where the hypotenuse, and the edge of the circle intersect. 

Using the same logic, it can be seen the the  is equal to the side adjacent to angle x. Therefore,  is equal to the x value of the point created where the hypotenuse and the edge of the circle intersect.

This information can be used to find any point on the unit circle if the corresponding angle is known.