1. Let’s extend our knowledge of trigonometric functions…
…in Section 4.3a

2. Trigonometric Functions of Any Angle
(not just positive or acute angles)
A more “dynamic” view of angles:
Initial Side – beginning position of a ray (part of the angle)
Vertex – the endpoint of this ray
Terminal Side – final position of a ray, after being rotated about
the vertex
Measure of an Angle – a number that describes the amount of
rotation from the initial side to the terminal side of the angle
Positive angles are generated by counterclockwise rotations
and negative angles are generated by clockwise rotations.

3. Trigonometric Functions of Any Angle
(not just positive or acute angles)
terminal
side
terminal
side
initial side
initial side
A positive angle in
standard position
A negative angle in
standard position

4. Trigonometric Functions of Any Angle
(not just positive or acute angles)
Coterminal Angles – angles that have the same initial side and
the same terminal side, but have different measures.
Positive and negative
coterminal angles
Two positive
coterminal angles

5. Guided Practice
Find and draw a positive angle and a negative angle that are
coterminal with the given angle.
Note: there are infinitely many possible solutions…
Here are two:
(a)

6. Guided Practice
Find and draw a positive angle and a negative angle that are
coterminal with the given angle.
(b)
The diagram?
(c)
The diagram? I believe that the diagram should always come first!

7. Definition: Trigonometric Functions of any Angle
Let be any angle in standard position and let P(x, y) be any
point on the terminal side of the angle (except the origin). Let r
denote the distance from P to the origin ( ). Then
P(x,y) y r x

8. Guided Practice
Let be the acute angle in standard position whose terminal
side contains the point (5, 3). Find the six trig functions of .
The diagram?
P(5,3) 3 5

9. Guided Practice
Let be any angle in standard position whose terminal side
contains the point (–5, 3). Find the six trig functions of .
The diagram?
P(–5,3) 3 –5

10. Guided Practice Find the six trigonometric functions of .
First, let’s draw a diagram, complete with a reference triangle:
P(1,–1)

11. Whiteboard Practice
Find each of the following without a calculator by using ratios in
a reference triangle:
(a)
(b)
(c)