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 also helps with circles, finding angles and distances in many shapes, coding in 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.

We usually know another angle θ.

And we give names to each side:

• Adjacent is adjacent (next to) to the angle θ
• Opposite is opposite the angle θ
• the longest side is the Hypotenuse

Sine, Cosine and Tangent

Trigonometry can often find a missing side or angle in a triangle.

The special functions Sine, Cosine and Tangent help us!

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

For any angle "θ":

Sine Function:	sin(θ) = Opposite / Hypotenuse
Cosine Function:	cos(θ) = Adjacent / Hypotenuse
Tangent Function:	tan(θ) = Opposite / Adjacent

(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°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57...

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

Example: What is the missing length here?

• We know the Hypotenuse
• We want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse:

sin(45°) = OppositeHypotenuse

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

sin(45°) = 0.7071...

Now we know all of this:

0.7071... = Opposite20

A little bit of algebra now. First swap sides:

Opposite20 = 0.7071...

Then multiply both sides by 20 (the Hypotenuse length):

Opposite	= 0.7071... × 20
	= 14.14 (to 2 decimals)

Try Sin Cos and Tan!

Move the mouse around to see how different angles affect sine, cosine and tangent:

Notice that the sides can be positive or negative by the rules of Cartesian coordinates. This makes the sine, cosine and tangent change between positive and negative also.

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:

© 2015 MathsIsFun.com v 0.81

And now you know why trigonometry is also about circles!

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

A big part of Trigonometry is 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 right-angled triangles.
	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!

Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin  When a coin is tossed, there are two possible outcomes: heads (H) or tails (T) We say that the probability of the coin landing H is ½. And the probability of the coin landing T is ½.

Throwing Dice  When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6.

Probability 

In general:

Probability of an event happening = Number of ways it can happenTotal number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index.

Words

Some words have special meaning in Probability:

Experiment or Trial: an action where the result is uncertain.

Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.

Sample Space: all the possible outcomes of an experiment

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. As there are 4 Kings that is 4 different sample points.

Event: a single result of an experiment

Example Events:

• Getting a Tail when tossing a coin is an event
• Rolling a "5" is an event.

An event can include one or more possible outcomes:

• Choosing a "King" from a deck of cards (any of the 4 Kings) is an event
• Rolling an "even number" (2, 4 or 6) is also an event

The Sample Space is all possible outcomes. A Sample Point is just one possible outcome. And an Event can be one or more of the possible outcomes.

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

Each time Alex throws the 2 dice is an Experiment.

It is an Experiment because the result is uncertain.

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

The Sample Space is all possible outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

These are Alex's Results:

Experiment	Is it a Double?
{3,4}	No
{5,1}	No
{2,2}	Yes
{6,3}	No
...	...

After 100 Experiments, Alex has 19 "double" Events ... is that close to what you would expect?

Area of Plane Shapes

Area is the size of a surface!
Learn more about Area, or try the Area Calculator.

	Triangle Area = ½ × b × h b = base h = vertical height			Square Area = a2 a = length of side
	Rectangle Area = w × h w = width h = height			Parallelogram Area = b × h b = base h = vertical height
	Trapezoid (US) Trapezium (UK) Area = ½(a+b) × h h = vertical height			Circle  Area = π × r2 Circumference = 2 × π × r r = radius
	Ellipse Area = πab			Sector Area = ½ × r2 × θ  r = radius θ = angle in radians

Note: h is at right angles to b:

Example: What is the area of this rectangle?

The formula is:

Area = w × h
w = width
h = height

We know w = 5 and h = 3, so:

Area = 5 × 3 = 15

Example: What is the area of this circle?

Radius = r = 3

Area		= π × r2
		= π × 32
		= π × (3 × 3)
		= 3.14159... × 9
		= 28.27 (to 2 decimal places)

Example: What is the area of this triangle?

Height = h = 12

Base = b = 20

Area = ½ × b × h = ½ × 20 × 12 = 120

A harder example:

Example: Sam cuts grass at $0.10 per square meter

How much does Sam earn cutting this area:

Let's break the area into two parts:

Part A is a square:

Area of A = a² = 20m × 20m = 400m²

Part B is a triangle. Viewed sideways it has a base of 20m and a height of 14m.

Area of B = ½b × h = ½ × 20m × 14m = 140m²

So the total area is:

Area = Area of A + Area of B = 400m² + 140m² = 540m²

Sam earns $0.10 per square meter

Sam earns = $0.10 × 540m² = $54