Trigonometric Functions
Right-triangle definitions of trigonometry are elegant, but these definitions only apply to acute angles. To expand the definitions of the trigonometry functions to all angles, we’ll first need to carefully define the measure of an angle.
In trigonometry, we tend to think of an angle as created by a rotating ray. The beginning position of the ray is called the initial side (usually, the initial side coincides with the positive x-axis; angles with such an initial side are said to be in standard position). The post-rotation position of the ray is called the terminal side. The measure of the angle is a number which describes the amount of rotation, as described in the section on radian measure. If the rotation of the ray was clockwise, the angle measure is a negative number; if the rotation was counterclockwise, the angle measure is positive.
An angle 
in standard position creates a right triangle on the coordinate axes as follows:
- Choose a point P0=(x0, y0) on the terminal side of the angle.
- Draw a vertical line from P0 to the x-axis (note: this line is perpendicular to the x-axis).
- The resulting right triangle with vertices (x0, y0), (x0, 0), (0,0) is called the reference triangle for the angle
.
Terminal Side
Initial side
The vertical side of this triangle has length y0; the horizontal side has length x0; the hypotenuse is the distance from P0 to the origin, or
.
Of course, we may choose an infinite number of points P0 on the terminal side. However, the triangles created by these points are all similar right triangles; therefore, the ratios of the sides of the triangles will remain the same. Thus the trigonometric functions of the angles of the reference triangle do not depend on which P0 we choose. This gives us a logical connection between right-triangle trigonometry for acute angles and trigonometry for all angles:
Trigonometric Functions of Any Angle
Notice that 
are not defined when the terminal side of the angle is vertical; 
are not defined with the terminal side of the angle is horizontal.
Reference Angles and Reference Triangles
Consider the angle 
. The terminal side of this angle (ray CA) creates a reference triangle in the fourth quadrant; the angle of this triangle which is in standard position (vertex at C) must be 
. Thus we say 
is the reference angle for 
.
The reference triangle created by 
is identical to the one created by 
--except that the triangle is reflected across the x-axis. Thus
Using Reference Angles to Find Trigonometric Function Values of 
- Sketch
in standard position; choose a point (x,y) on the terminal side. - Construct the reference triangle (vertical side is the line segment from (x,y) to (x,0); horizontal side is the line segment from (x,0) to (0,0); hypotenuse is the line segment connecting (0,0) and (x,y).
- The reference triangle contains an acute angle
(its vertex is at the origin) coterminal to 
(in quadrant I), 
(in quadrant II), 
(in quadrant III) or 
(in quadrant IV). Right triangle trigonometry will determine the trigonometric function values of 
. - The trigonometric function values of
are the absolute value of the trigonometric function values of 
. Use the position of the reference triangle to determine the signs for 
.
When P0 is on the Unit Circle
If the distance from P0 =(x0, y0) to the origin is 1, then 
. Therefore, every point on the unit circle can be written as 
, where 
is the central angle in standard position whose terminal side intercepts the unit circle at P0.
Because the definitions of the trigonometric functions are so simple when P0 is on the unit circle, we often pick P0=(cos(
), sin(
)) when we’re calculating the trigonometric function values of 
.
Coterminal Angles
Since every point on the unit circle can be written as 
, and one rotation through a circle is defined to be 
radians, the angles 
and 
(where k is an integer) must have terminal sides in identical positions. We call such angles coterminal; from the definitions we’ve given, coterminal angles must have the same trigonometric function values.
Angles 
and 
are coterminal.
Quadrantal Angles
Angles such that the terminal side of the angle lies on one of the coordinate axes are called the quadrantal angles.
If the terminal side coincides with the x axis (so 
for some integer k), the distance from the origin is the absolute value of the x-coordinate while the y-coordinate is 0. By the rules above, then, sin(
)=tan(
)=0, cos(2
)=sec(2
)=1, cos(
)=sec(
)= -1. Note that we cannot define the cotangent or cosecant of a quadrantal angle which coincides with the x axis.
If the terminal side coincides with the y axis (so 
for some integer k), the distance from the origin is the absolute value of the y-coordinate while the x-coordinate is 0. Again, the rules above give sin(
)=csc(
)=1, sin(
)=csc(
)=-1, cos(
)=cot(
)=0. Note that we cannot define the tangent or secant of a quadrantal angle which coincides with the y axis.
Quadrantal angles are general considered “special angles,” like 
, 
, or 
. You should memorize the trigonometric function values at the quadrantal angles.
