Notes on quadrants

1. Topic 2-4Topic 2-4
Trig FunctionsTrig Functions

2. Learning GoalLearning Goal
To determine the values of the six To determine the values of the six
trigonometric ratios in a right triangle.trigonometric ratios in a right triangle.

3. Defining Trig Functions
r
θ
Let there be a point P (x, y) on a
coordinate plane.
r is the distance from the origin
to the point P, which can be
represented as being on the
terminal side of θ. Since r
represents a distance, it is
always positive.
P(x, y)

4. Trig Functions
For a given value of θ, the ratios of y/r, x/r and y/x are constant
because of similar triangles, thereby satisfying the requirement of a
function that for each x-value (or, in this case, θ) there is only one y-
value.
r
y
sin
y
r
csc
r
x
cos
x
r
sec
x
y
tan y
x
cot
sin
cos



1
sin

1
cos

1cos
tansin


 

5. Finding Trig Function
Values - 1
If the terminal side of θ in standard position passes through
point P (6, 8), draw θ and find the exact value of the six trig
functions of θ.
P(6, 8)
θ
8
6
r = 10
84
sin
105
y
r

63
cos
105
x
r

84
tan
63
y
x

105
csc
84
r
y

105
sec
63
r
x

63
cot
84
x
y


6. Finding Trig Function
Values - 2
If the terminal side of θ in standard position passes through
point P (-4, 2), draw θ and find the exact value of the six trig
functions of θ.
P(-4, 2)
θ
2
-4
r = 2√5
2 5
sin
525
y
r
 
4 25
cos
525
x
r

-
 -
2 1
tan
42
y
x
-
-
25
csc 5
2
r
y
 
25 5
sec
4 2
r
x
 -
-
4
cot 2
2
x
y

-
-

7. Positive Trig Function
Values
r
r
r
r
x-x
y
y
-y -y
All trig functions
are positive.
Sine is positive.
(So is cosecant.)
Tangent is positive.
(So is cotangent.)
Cosine is positive.
(So is secant.)

8. Determining The
Quadrant - 3
In which quadrant is θ if cos θ and tan θ have the same sign?
Quadrants I and II

9. Determining The
Quadrant - 4
In which quadrant is θ if cos θ is negative and sin θ is positive?
Quadrant II

10. Determining The
Quadrant - 5
In which quadrant is θ if cot θ and sec θ have opposite signs?
Quadrants III and IV

11. Find values - 1
Find the exact values of the other five trig functions for an
angle θ in standard position, given
53
sin , 2
132
p
 p- <<
3π/2
2π
13
-5
12
12
cos
13

13
csc
5
-
13
sec
12

5
tan
12
-
12
cot
5
-
θ

12. Find values - 2
Find the exact values of the other five trig functions for an
angle θ in standard position, given
sec 2, 2 pp- <<
π
2π
2
-1
3
sin
2
-
1
cos
2
-
2 23
csc
33
--
tan 3
1 3
cot
33

θ
-√3