## What is Trigonometry?

Just the word *trigonometry* makes you think of **triangles**. Trigonometry is the **study of the measure of the angles and sides of triangles**.

## Triangle Review

Triangles come in different shapes and sizes, but they all have one thing in common, their **three interior angles sum to 180 degrees**. Also, the sum of two side lengths of a triangle must be greater than the length of the third side.

The first example is an equilateral triangle, where all interior angles are 60 degrees:

Next is an image of an isosceles triangle, which has two angles of the same degree:

Here you can see a scalene triangle, where all sides have different lengths and all angles are different:

The last example image shows a right triangle, which has one interior angle of 90 degrees:

### Pythagorean Theorem

More than 2,000 years ago, Greek mathematician **Pythagoras** made a discovery about the relationship of the sides of right triangles. Today, we know this discovery as the **Pythagorean Theorem**. For all **right triangles**, with the hypotenuse as side c, the Pythagorean Theorem states:

$a^{2} + b^{2} = c^{2}$

## Trigonometric Ratios in Right Triangles

For all **right triangles**, there is also a special relationship between the three sides and the three angles. The relationship between a triangle’s sides and angles is the basis of trigonometric ratios.

### Finding Trigonometric Ratios

To study the **trig ratios**, you need to know how the appropriated terminology to describe the sides and their related angles.

The **hypotenuse** is located opposite the right angle. For angle A, the side across from the angle is the **opposite side**, and the side next to the angle, but not the hypotenuse, is the **adjacent side**.

The basic trig ratios are sine, cosine, and tangent. An easy mnemonic device to remember the trigonometric ratios is **SOHCAHTOA**:

### Using Trigonometric Ratios to Find Angles

To find unknown angles, use the trig ratios to set up an equation then solve for the unknown value.

To use the **sine ratio** to solve for the unknown angle, use the trig chart or use a calculator to find the **inverse of sine x**.

**Example:**
Find the measure of angle x.

$\begin{align} \sin x &= \frac{opp}{hyp}\\ \sin x &= \frac{6}{10}\\ \sin x &= 0.6\\ x &= \frac{0.6}{sin}\\ x &= 36.869 \end{align}$

x is approximately 37 degrees

### Using Trigonometric Ratios to Find Distances

To find unknown sides, use the trig ratios to set up an equation then solve.

To use the **sine ratio** to solve for the unknown side, use the trig chart or a calculator.

**Example:**
Find the length of side x.

$\begin{align} \sin 53 &= \frac{x}{5}\\ 0.8&=\frac{x}{5}\\ 0.8 \times 5&=\frac{x}{5}\times 5\\ x&=4.0 \end{align}$

The length of the unknown side is 4 units.

## Special Triangles

There are two **special triangles**, **45-45-90** and **30-60-90**. What’s special about these two triangles? The angles and side lengths of the two special triangles have a *special* pattern that holds for any size triangle with the same angles.