Trigonometric functions
Definitions using right triangles
The trigonometric functions can be defined using a right triangle.
The definitions are
and they tell how the ratios of the different sides relate to the angle.
A problem with this definition occurs whenever the angle .
Definition using the unit circle
The remedy for the problem is to define the trigonometric functions using the unit circle.
Let be the point on the unit circle at the angle .
is defined as the -coordinate of .
is defined as the -coordinate of .
is defined as the ratio between and .
The strength of this definition is that is not restricted to the interval , but can take on all real values for and .
For we have a clear definition for all values of except when , where is a whole number. This is because we can not divide by and for those will be zero.
Degrees and radians
Degrees
Degrees are a unit used to measure angles. is one full rotation around a circle, and one degree is thus turn around a circle as shown in the following picture.
Radians
Radians are, just as degrees, a unit to measure angles. However, instead of one full turn around a circle being , it's : the circumference of a circle! In fact an angle of one radian is a turn around a circle by exactly one radius as shown in the picture below.
Conversion
The conversion of an angle from degrees to radians is given by the following formula
and the analogous formula for converting in radians to degrees is
When referring to angles we generally use radians, except when otherwise specified. It is therefore recommended to get comfortable with radians as early as possible.
Trigonometric rules
According to this image, the -coordinate corresponds to the value of and the -coordinate corresponds to the value of . Remember:
and:
Some useful and important identities to remember are:
