The CAST Diagram
This diagram shows the
following:
 Between 0 and 90
degrees, all the
graphs are
positive.
 Between 90 and
180 degrees, only
sine is positive.
 Between 180 and
270 degrees, only
tangent is positive.
 Between 270 and
360 degrees, only
cosine is positive.
When we try and solve a trigonometric equation, we can use a calculator
to find one solution.
Example:
sin x = 0.4
=>
x = sin-1 (0.4) = 23.58
Remember the sine wave, its shape means that it takes on the value 0.4 at
a second position.
Checking the CAST diagram, we can see that this second position must
be in Quadrant 2 as this is the only other place where sine is positive.
To find an equivalent value in Quadrant 2, we must find
180 – 23.58 = 156.42
Any further solutions can be found by adding 360 to both of the solutions,
this is due to the periodicity of the graph.
Questions:
Find values for x such that 0≤x≤360
a)
b)
c)
sin x = 0.7
sin x = 0.2
sin x = 0.9
Example 2:
cos x = 0.6
=>
x = cos-1 (0.6) = 53.13
Remember the cosine wave, its shape means that it takes on the value 0.6
at a second position.
Checking the CAST diagram, we can see that this second position must
be in Quadrant 4 as this is the only other place where cosine is positive.
To find an equivalent value in Quadrant 4, we must find
360 – 53.13 = 306.87
Any further solutions can be found by adding 360 to both of the solutions,
this is due to the periodicity of the graph.
Questions:
Find values for x such that 0≤x≤360
a)
b)
c)
cos x = 0.64
cos x = 0.13
cos x = 0.55
Example 3:
tan x = 2.7
=>
x = tan-1 (2.7) = 69.68
The tangent graph takes on the value 2.7 in a second position.
Checking the CAST diagram, we can see that this second position must
be in Quadrant 3 as this is the only other place where tangent is positive.
To find the equivalent value in Quadrant 3, we must find
180 + 69.68 = 249.68
Any further solutions can be found by adding 180 to both of the solutions,
this is due to the periodicity of the graph.
Questions:
Find values for x such that 0≤x≤360
a)
b)
c)
tan x = 3.1
tan x = 0.4
tan x = 6.2
Mixed Questions:
Find values for x such that 0≤x≤360
a)
c)
e)
sin x = 0.43
cos x = 0.6
tan x = 5.7
b)
d)
f)
tan x = 5.2
sin x = 0.14
cos x = 0.56