Transcript Trig Graphs & The CAST diagram

Trig Graphs & The CAST diagram
Tuesday, 21 July 2015
The graph of
y  sin x
y
1
0.5
0
-0.5
-1
180
360
x
The graph of
y  cos x
y
1
0.5
0
-0.5
-1
100
200
300
x
The graph of
y
y  tan x
20
10
0
-10
-20
180
360
x
The CAST Diagram
Sin only
positive
2nd
Quadrant
3rd
Tan only
positive
1st
Quadrant
4th
Quadrant Quadrant
On the CAST diagram All angles
are measured Anti-Clockwise
All positive
Cos only
positive
Example
Given that  lies between 0° and 360° and that sin 𝜃 =
1
2
indicate
possible values of  on separate diagrams. State the possible values of .
y
From the graph we can see
that there are two solutions
1
On the CAST diagram there are two possible
angles. One acute the other obtuse.
1
x  sin 1  
2
A
S
 30
0
A
S
30°
30°
T
C
x = 150°
0.5
T
-1
C
x = 30°
-0.5
180
360
x
Example
Given that  lies between 0° and 360° and that tan 𝜃 = − 3 indicate
possible values of  on separate diagrams. Statey the possible values of .
20
From the graph we can see
that there are two solutions
10
On the CAST diagram there are two possible
angles. One reflex the other obtuse.
1

0

180
360
x  tan  3  60
-10
A
S
A
S
-20
60°
60°
T
C
x = 120°
T
C
x = 300°
On the CAST diagram A
negative angle means we
move Clockwise
x
Example
Solve each of the following equations for 0 ≤ 𝑥 ≤ 360°, giving your
answers correct to 2 decimal places
(i) 3 sin 𝑥 = 2
(ii) 1 − 4 cos 𝑥 = 0
(iii) 5 + 6 tan 𝑥 = 1
(i)
2
3
x  41 .8
sin x 
A
S
41.8°
41.8°
T
C
x = 138.2°
A
S
T
C
x = 41.8°
1
(ii) cos x 
4
x  75 .5
A
S
A
S
75.5°
75.5°
T
C
T
x = 284.5°
4
6
x  33.7
C
x = 75.5°
(iii) tan x  
A
S
A
S
33.7°
33.7°
T
C
x = 146.3°
T
C
x = 326.3°
Example
Solve the equation cos 2 𝑥 =
cos x  
1
2
1
cos x 
2
x  45
1
cos
x


(i)
2
x  135
1
2
in the range 0 ≤ 𝑥 ≤ 360°.
A
S
A
S
45°
45°
T
C
T
C
x = 315°
x = 45°
A
S
A
S
135°
T
C
x = 135°
T
135°
x = 225°
C
Trig Equations
1. Solve each of the following equations in the range of 0º to 360º
1
a) cos x 
2
b) sin x  0.4
c) tan x  1.4
d) 4 sin x  3
e) 9 cos x  7
f) 1  2 sin x  0
g) 1  5 tan x  0
2. Solve the equation sin 2 x  0.3 in the range 0º to 360º