Section 7.2 - Right Triangle Trig

Section Objectives

  1. Use right triangles to define and evaluate the trig functions.
  2. Recognize and use basic trig identities.
  3. Use trig to solve right triangles.
  4. Use (and memorize) the values of the trig functions at , , and .


Right Triangle Approach to the Trig Functions

Let be an acute angle of a right triangle. We can define the trigonometric functions at in terms of the lengths of sides of the right triangle.


There are six basic trigonometric functions: sine (), cosine (), tangent (), secant (), cosecant (), and cotangent ().

Right triangle definitions of the trig functions


Trig students often remember the right triangle definitions of the sine, cosine, and tangent by using SOH CAH TOA.



Examples

Let be the smallest angle of the 5-12-13 right triangle. Sketch the triangle and find the values of all six trig functions at .





Think back to your geometry class when you studied the -- triangle with legs of length 1. Sketch the triangle and use it to find the values of all six trig functions at .





Think back to your geometry class when you studied the -- triangle with sides of length 1, , and 2. Sketch the triangle and use it to find the values of all six trig functions at and .





Confunction Identities

Because the acute angles of a right triangle are complementary, it is easy to understand the following cofunction identities.

 



Some Other Trig Identities

Trigonometric identities can be extremely useful. Here are a few basic identities that follow immediately from the definitions:



These next identities are called Pythagorean identities.




Examples

Show that




Show that




At 19 feet from the base of a flagpole, the angle of elevation to the top is . How tall is the flagpole?




Preview of the next section...

Sketch the unit circle and a 1st quadrant angle, . The sine and cosine of are easily described in terms of the coordinates of the intersection point of the terminal side and the circle. This is unit circle trigonometry.

Unit circle trig