Animated Unit Circle
Pick an angle and then draw a line from the origin rotated from the positive x-Axis. This line intersects the unit circle at a point .
Then we define and .
Next draw a line tangent to the circle at this will intersect the x-Axis at a point and it will interest the y-Axis at .
We then define the following functions: .
Caveat that and are signed length with them being positive when have the same sign and negative otherwise. This is because is defined as and this is better seen with another (technically ore accurate) rearrangement of the unit circle, but that one does not as clearly show the idea of co- functions and how they are related to their counterpart.
Notice that this forms similarright-angled triangles, one with sides with perpendicular being . (call this triangle )
Another with sides and perpendicular height . This gives rise to the names of the functions as they represent the same thing but on the triangle with angle at the origin vs. the triangle with angle at the origin. (we will call this “co” angle, and call the triangle )
Finally a smaller triangle triangle can be formed inside the unit circle with radius as hypotenuse with other lengths . (call this triangle )
Identities
Using these triangles we can derive important relations.
First using similarity of and we derive that .
We can also derive the reciprocal relationships
Using Pythagoras we can also derive the following
Function Properties and Graphs
Domain of
Range of
Period of radians
Domain , This restricted domain exists because has no solutions for
Range is , This range is due to the one-to-one mapping to only the principal root to ensure the function exists
Domain of
Range of
Period of radians
Domain , This restricted domain exists because has no solutions for
Range is , This range is due to the one-to-one mapping to only the principal root to ensure the function exists
Domain
Range
Period of radians
Domain
Range
Non-Periodic
Domain
Range of
Period of radians
Domain
Range of
Period of radians
Domain
Range
Period of radians
Addition Formulae
Inside Addition
Outside Addition
T-Formulae
Definitions/Equations
The t-formulae are formulae for the trigonometric functions using the substitution to obtain the following formulae:
Solving Equations
To solve questions, you need to find if not directly given, then use the t-formula for what you’re trying to find. You generally find
Finding or it will either be given in the question or the reciprocal function Then using the identity to obtain the other function Either sine or cosine then choose the correct sign for the square root based on boundary conditions for in the problem. Then use
Weierstrass Substitution For Integrals
You can use the t-formulae as a substitution for integrals it should be self explanatory from there but a key fact to remember is
Pick an angle 
