Download 13-3A Trig Functions of General Angles

Algebra 2
13.3A Trigonometric Functions of General Angles
Obj: able to find values of trigonometric functions for general angles; able to use reference triangles
Evaluating Trigonometric Functions
Let θ (Theta) be an angle in standard position with any point (x, y) (except the origin)
on the terminal side of
θ.
Let r =
y
(x, y)
x +y .
2
2
y
y
x
tan θ = , x ≠ 0
cosθ =
r
x
r
r
x
r
csc θ = , y ≠ 0
cot θ = , y ≠ 0
sec θ = , x ≠ 0
y
y
x
Remember: (x, y) can have positive or negative values. For acute angles, the
definitions above give the same values as those given by the definitions in Lesson 13.1
sin θ =
r
y
θ
x
x
Initial side
r = x2 + y 2
The Reference Angle of an Angle
Let θ be an angle in standard position. Its Reference Angle is the acute angle θ ' (read theta prime) formed by the terminal side
of θ and the x-axis.
(x, y)
Quadrant IV
Quadrant III
Quadrant II
y
y
y
θ
θ
θ
Initial side
x
x
θ‘
θ‘
Initial side
θ‘
Initial side
(x, y)
(x, y)
Evaluating Trigonometric Functions of Any Angle
Use the following steps to evaluate a trigonometric function of any angle θ .
1. Find the reference angle θ ’.
2. Evaluate the trigonometric function for the angle θ ’.
3. Use the quadrant in which θ lies to determine the sign of the trigonometric function of
Find the exact value of the six trigonometric functions of
1. (-5, 12)
θ
x
if the terminal side of
2. ( -3, -4)
θ
θ.
in standard position contains the point.
Find the exact value of the six trigonometric functions of
3. (0, 3)
I don’t get
it at all
1
What do you still need to work on?
I sort of
get it
2
θ
if the terminal side of
4. ( 5, -5)
θ
in standard position contains the point.
Rate yourself on how well you understood this lesson.
I understand
I understand
most of it but I
it pretty well
need more practice
3
4
I got it!
5