1. Introduction to Trigonometry
Trigonometry (from Greek trigonon "triangle" + metron "measure")
Want to Learn Trigonometry?
Here is a quick summary.
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.

2. Angles
Angles (such as the angle "θ" above) can be in Degrees or Radians. Here are
some examples:
Angle
Degrees
Radians
90°
π/2
__ Straight Angle
180°
π
Full Rotation
360°
2π
Right Angle
"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":
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°?

3. 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 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, it is easy to measure
sine, cosine and tangent.
Here you can see the sine function being made by the unit circle:
You can 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.

4. 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)
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. "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:

5. If you know any 3 of the sides or angles you 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 / Opposit
Trigonometric Identities
Right Triangle
The Trigonometric Identities are equations that are true for Right Angled
Triangles ...... if it is not a Right Angled Triangle refer to our Triangle
Identities page.
Each side of a right triangle has a name:

6. (Adjacent is adjacent to the angle, and Opposite is opposite ... of course!)
Important: We are soon going to be playing with all sorts of functions and it can
get quite complex, but remember it all comes back to that simple triangle with:




Angle θ
Hypotenuse
Adjacent
Opposite
Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent.
They are just the length of one side divided by another
For a right triangle with an angle θ :
Sine Function: sin(θ) = Opposite / Hypotenuse
Cosine Function: cos(θ) = Adjacent / Hypotenuse
Tangent Function: tan(θ) = Opposite / Adjacent
Also, if we divide Sine by Cosine we get:

7. So we can also say:
tan(θ) = sin(θ)/cos(θ)
That is our first Trigonometric Identity.
But Wait ... There is More!
We can also divide "the other way around" (such
as Adjacent/Opposite instead ofOpposite/Adjacent):
Cosecant Function: csc(θ) = Hypotenuse / Opposite
Secant Function: sec(θ) = Hypotenuse / Adjacent
Cotangent Function: cot(θ) = Adjacent / Opposite
Example: if Opposite = 2 and Hypotenuse = 4 then
sin(θ) = 2/4, and csc(θ) = 4/2
Because of all that we can say:
sin(θ) = 1/csc(θ)
cos(θ) = 1/sec(θ)
tan(θ) = 1/cot(θ)
And the other way around:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
And we also have: cot(θ) = cos(θ)/sin(θ)
cot(θ) = 1/tan(θ)

8. More Identitites
There are many more identities ... here are some of the more useful ones:
Opposite Angle Identities
sin (-θ) = - sin (θ)
cos (-θ) = cos (θ)
tan (-θ) = - tan (θ)
Double Angle Identities
Half Angle Identities
Note that "±" means it may be either one, depending on the value of θ/2

9. Angle Sum and Difference Identities
Note that
means you can use plus or minus, and the
opposite sign.
means to use the