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

triangle Trigonometry ... is all about triangles.

It also helps with circles, finding angles and distances in many shapes, coding in video games, and more!

Right Angled Triangle

triangle showing Opposite, Adjacent and Hypotenuse

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:

Sine, Cosine and Tangent

Trigonometry is good at 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":

Right-Angled Triangle

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°?

triangle 2.8 4.0 4.9 has 35 degree angle

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:

right angle triangle 45 degrees, hypotenuse 20

Example: What is the missing length here?

Sine is the ratio of Opposite / Hypotenuse:

sin(45°) = Opposite Hypotenuse

calculator-sin-cos-tan

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

sin(45°) = 0.7071...

Now we know all of this:

0.7071... = Opposite 20

A little bit of algebra now. First swap sides:

Opposite 20 = 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

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:

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 angleRight Angle  90° π/2
__ Straight Angle 180° π
right angle 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"

trig ASA example

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:

Right-Angled Triangle

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:

right angled triangle

The Trigonometric Identities are equations that are true for all right-angled triangles.

triangle

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!