Introduction to Trigonometry

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

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Here is a quick summary.
Follow the links for more, or go to Trigonometry Index

 

triangle Trigonometry ... is all about triangles.

Right Angled Triangle

triangle showing Opposite, Adjacent and Hypotenuse

A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side:

  • Adjacent is adjacent to the angle "θ",
  • Opposite is opposite the angle, and
  • the longest side is the Hypotenuse.

Angles

Angles (such as the angle "θ" above) 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

"Sine, Cosine and Tangent"

The three most common functions in trigonometry are Sine, Cosine and Tangent. We will use them a lot!

They are simply one side of a 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

 

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...

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

 

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Try It!

Have a try! Move the mouse around to see how different angles affect sine, cosine and tangent

And you will also see why trigonometry is also about circles!

Notice that the sides can be positive or negative according to the rules of Cartesian coordinates. This makes the sine, cosine and tangent vary 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:

You can also see the nice graphs made by sine, cosine and tangent.

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation.

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

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°

It is also possible to find missing side lengths and more. 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

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!