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?

Why is this triangle so important?
Imagine we can measure along and up but want to know the direct distance and angle:

Or 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°) = OppositeHypotenuse = 2.84.9 = 0.57...

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

Example: What is the missing height 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...

0.7071... is the ratio of the side lengths: in other words the Opposite is about 0.7071 times as long as the Hypotenuse.
Maybe you can figure out the height now?
But let's do it formally using some algebra:

Start with:sin(45°) = OppositeHypotenuse

Put in what we know:0.7071... = Opposite20

Swap sides:Opposite20 = 0.7071...

Multiply both sides by 20:Opposite = 0.7071... × 20

Calculate:Opposite = 14.14 (to 2 decimals)

Done!

Try Sin Cos and Tan

What are Real Numbers?

When both the rational and irrational numbers combined, the combination is defined as the real numbers. Real numbers can be both positive or negative, and they are denoted by the symbol “R”. Numbers like a natural number, decimals, and fraction comes under the real number.

Classification of Real Numbers

• Natural Numbers– It includes all the counting numbers such as 1, 2, 3, 4,…
• Whole Numbers– Numbers starting with zero are called whole numbers, like 0, 1, 2, 3, 4,…
• Integers– Whole numbers and negative of all natural numbers are collectively known as integers, for example -3, -2, -1, 0, 1, 2,
• Rational Numbers– All the numbers that can be written in the form of p/q, where $q\ne 0$ are known as Rational numbers.
• Irrational Numbers– The numbers which cannot be written in the form of p/q (simple fraction) are known as irrational numbers. Irrational numbers are non-terminating and non-repeating in nature.

Properties of real numbers

• Commutative property- If we have real numbers m,n. The general form will be m + n = n + m  for adaddition andmn = nm for multiplication
• Associative property- If we have real numbers m,n,r. The general form will be  m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication
• Distributive property- If we have real numbers m,n,r. The general form will be – m (n + r) = mn + mr and (m + n) r = mr + nr
• Identity property- For addition: m + (- m) = 0