What is Trigonometry?

Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying.

Trigonometry ... is all about triangles.

Right Angled Triangle

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 Angle	90°	π/2
__ Straight Angle	180°	π
Full Rotation	360°	2π

"Sine, Cosine and Tangent"

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

They are simply one side of a 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

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 abbreivated to sin, cos and tan.

Unit Circle What we have just been playing with is the Unit Circle. It is just a circle with a radius of 1 with its center at 0. Because the radius is 1, it is easy to measure 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 you need to calculate the function for an angle larger than a full rotation of 2π (360°) just subtract as many full rotations as you need 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)

Likewise if 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. By "solving" I mean finding missing sides and angles.

Example: Find the Missing Angle "C"

It's easy to find angle C by 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:

If you know any 3 of the sides or angles you can find the other 3
(except for the three angles case)

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

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!

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