# Introduction to Trigonometry

**Trigonometry** (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary.

Follow the links for more, or go to Trigonometry Index

Trigonometry ... is all about triangles. |

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

## Right-Angled Triangle

The triangle of most interest is the right-angled triangle. The right angle is shown by the little box in the corner:

Another angle is often labeled θ, and the three sides are then called:

**Adjacent**: adjacent (next to) the angle θ**Opposite**: opposite the angle θ- and the longest side is the
**Hypotenuse**

### Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

## Sine, Cosine and Tangent

The main functions in trigonometry are ** **Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle "** θ**":

*(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)*

### Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = \frac{Opposite}{Hypotenuse} = \frac{2.8}{4.9} = **0.57...**

The triangle could be larger, smaller or turned around, but **that angle will always have that ratio**.

Calculators have sin, cos and tan to help us, so let's see how to use them:

### Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

- We know the
**Hypotenuse** - And we want to know the
**Opposite**

**Sine** is the ratio of **Opposite / Hypotenuse**:

sin(45°) = \frac{Opposite}{Hypotenuse}

Get a calculator, type in "45", then the "sin" key:

sin(45°) = **0.7071...**

What does the **0.7071...** mean? It is the ratio of the side lengths, so the Opposite is *about 0.7071* times as long as the Hypotenuse.

We can now put **0.7071...** in place of sin(45°):

**0.7071...** = \frac{Opposite}{Hypotenuse}

And we also know the hypotenuse is **20**:

0.7071... = \frac{Opposite}{20}

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Finally:

Opposite = **14.14m** (to 2 decimals)

### Example: How Tall is The Tree?

**20**: Opposite = 0.7071... × 20

**14.14**(to 2 decimals)

The tree is 14.14m tall

## Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be **positive or negative** by the rules of Cartesian coordinates, so the sine, cosine and tangent change between positive and negative also.

So **trigonometry is also about circles**!

## Unit Circle

What you just played with is the Unit Circle.

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

*Note: you can see the nice graphs made by sine, cosine and tangent.*

## Degrees and Radians

Angles can be in Degrees or Radians. Here are some examples:

Angle | Degrees | Radians |
---|---|---|

Right Angle | 90° | π/2 |

__ Straight Angle | 180° | π |

Full Rotation | 360° | 2π |

## Repeating Pattern

Because the angle is **rotating around and around the circle** the Sine, Cosine and Tangent functions **repeat once every full rotation** (see Amplitude, Period, Phase Shift and Frequency).

When we want to calculate the function for an angle larger than a full rotation of 360° (2π radians) we subtract as many full rotations as needed to bring it back below 360° (2π radians):

### Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = **0.985 **(to 3 decimal places)

And when the angle is less than zero, just add full rotations.

### Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2π radians

−3 + 2π = −3 + 6.283... = 3.283... radians

sin(−3) = sin(3.283...) = **−0.141 **(to 3 decimal places)

## Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles. "Solving" means finding missing sides and angles.

### Example: Find the Missing Angle "C"

Angle **C** can be found using angles of a triangle add to 180°:

So C = 180° − 76° − 34° = **70°**

We can also find missing side lengths. The general rule is:

**When we know any 3 of the sides or angles we can find the other 3**

(except for the three angles case)

See Solving Triangles for more details.

## Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other **trigonometric functions** which are made by dividing one side by another:

Cosecant Function: | csc(θ) = Hypotenuse / Opposite |

Secant Function: | sec(θ) = Hypotenuse / Adjacent |

Cotangent Function: | cot(θ) = Adjacent / Opposite |

## Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

The Trigonometric Identities are equations that are true for all | |

The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle). |

Enjoy becoming a triangle (and circle) expert!