Algebra & Trig, Sullivan & Sullivan
Fourth Edition
§6.4
Page 1 / 2
§6.4 – Trig Functions of General Angles
Up till now, we've (technical) been defining all our
trig functions based on the angles inside a right triangle.
We'd like to generalize these functions, and be able to
define them on any angle. So we'll build on the intuition
we've got from triangles, and (re) define trig functions
more generally.
In a nutshell, we're going to define (acute) triangles per quadrant, for all four quadrants, and
thus be able to set θ to any degree between 0 & 360 (or any multiple thereof). We'll start by
looking at the sides (more or less ignoring θ for now), then come back to θ later.
Trig Functions of General Angles
Let θ be any angle (in standard position), and let (x,y) be a point on the terminal line (other than
(0,0)).
Clearly, the distance to that point is dist  x 2  y 2 . This is also the length of the hypotenuse.
X is the length of the adjacent side, y is the length of the opposite side.
We can then define the 6 trig functions as:
Sine:
Cosine:
Opposite
Sin 
Hypotenuse
Cosecant:
Hypotenuse
Csc 
Opposite
Adjacent
Cos 
Hypotenuse
Secant:
Hypotenuse
Sec 
Adjacent
Tangent:
Opposite
Tan 
Adjacent
Cotangent:
Adjacent
Cot 
Opposite
(Looks familiar, don't it?  )
MAJOR POINT: Now, unlike before, we can have negative values for x & y, so we might end up
with trig functions that produce negative values!
MAJOR POINT: If we pick a quandrantal angle, one of the sides of the triangle will end up
being 0, which might lead to the trig function being 0 overall (if the zero is in the numerator), or
might lead to the trig function being undefined overall (if the zero is in the demoninator)(since
we can't divide by zero)
This is all well & good, and allows us to define trig functions for 0 ≤ θ ≤ 360
Algebra & Trig, Sullivan & Sullivan
Fourth Edition
§6.4
Page 2 / 2
+/- Signs of Trig Functions
This is another variation on the 'we'll give you some info, but not all of it, and you find the
remaining info'
Good practice for 'can you sleuth out the answer?' sorts of problems.
II
Sin/Csc > 0
All others negative
I
All Trig Positive
III
Tan/Cot > 0
All others negative
IV
Cos/Sec > 0
All others negative
Coterminal Angles
Coterminal angles – two angles that (when drawn in standard position) end up putting their
teriminal sides in the exact same place.
We can use this to avoid having to define trig functions for greater than 360° - we simply
subtract 360 till we find something that's only got one rotation
Similarly, we can add 360 to a negative angle until it's within 0 and -360
Ex: Tan 420°: 420° – 360°  60°, so it's the same as Tan 60°
Reference Angles
Reference angles – Similar idea to coterminal angles – we're looking to simplify how we deal
with these trig functions. A reference angle is the angle between the terminal side, and the
nearest side (positive or negative) of the X axis.
Ex: Tan 420°: 420° – 360°  60°, so it's the same as Tan 60°