# Section 1.3 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.

### Right Triangle Approach to the Trig Functions

While focusing on the 1st quadrant, let $\theta$ be an acute angle and look back at our unit circle approach to the trig functions. All along, there has been a right triangle almost in plain view.

Now ignore everything except that right triangle, and suppose that the hypotenuse no longer is required to have length 1. The trig functions can then be defined on the right triangle.

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

#### Examples

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

Think back to your geometry class when you studied the $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle with legs of length 1. Sketch the triangle and use it to find the values of all six trig functions at $\pi/4$.

Think back to your geometry class when you studied the $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle with sides of length 1, $\sqrt{3}$, and 2. Sketch the triangle and use it to find the values of all six trig functions at $\pi/6$ and $\pi/3$.

### Some Trig Identities

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

$\displaystyle \sin \theta = \frac{1}{\csc \theta} \qquad \qquad \csc \theta = \frac{1}{\sin \theta}$

$\displaystyle \cos \theta = \frac{1}{\sec \theta} \qquad \qquad \sec \theta = \frac{1}{\cos \theta}$

$\displaystyle \tan \theta = \frac{1}{\cot \theta} \qquad \qquad \cot \theta = \frac{1}{\tan \theta}$

$\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta} \qquad \qquad \cot \theta = \frac{\cos \theta}{\sin \theta}$

These next identities are called Pythagorean identities.

$\sin^2 \theta + \cos ^2 \theta = 1$

$\tan^2 \theta + 1 = \sec^2 \theta$

$1 + \cot^2 \theta = \csc^2 \theta$

We have already discussed the first Pythagorean identity. The others follow from basic identities.

#### Examples

Show that $\sin \theta \, \csc \theta = 1$

Show that $(\csc \theta + \cot \theta)(\csc \theta - \cot \theta)=1$

At 19 feet from the base of a flagpole, the angle of elevation to the top is $64.6^{\circ}$. How tall is the flagpole?