(e)
Solve for x: (-Sin(x)) x (Cos(2x)) = 1 for x ∈ [-180 ; +180]
Use Cos(2x)
(-Sin(x)) x (Cos(2x))
-Sin(x) x (1 – 2Sin2x) - 1
+2Sin3x -Sin(x) - 1
=
=
=
=
1 – 2Sin2x
-Sin(x) x (1 – Sin2x) =
0
0
1
By trial and error Sin(x) = 1 is a solution
(Sin(x) – 1) )+2Sin3x + 0x - Sin(x)
- 1 ( 2Sin2x - 2Sin(x) + 1
3
2
2Sin x - 2Sin x
----------------------*
2Sin2x - Sin(x)
2Sin2x – 2Sin(x)
-------------------*
Sin(x) - 1
Sin(x) - 1
--------------*
*
(Sin(x) – 1)(2Sin2x - 2Sin(x) + 1)
and
=
0
2Sin2x - 2Sin(x) + 1 has no real roots and so
(Sin(x) – 1) = 0  Sin(x) = 1  x = 90 + 360k is the only solution.
See graph over page.
(f)
Find the values of x for which: (-Sin(x)) x (Cos(2x)) > 0 for x ∈ [0 ; +180]
+2Sin3x
-Sin(x)
>
0
Sin(x) (2Sin2(x) - 1)
>
0
Critical values when Sin(x) = 0
OR
2Sin2(x) - 1 = 0  2Sin2(x) = 1
Remember: x ∈ [0 ; +180]
Sin(x) = 0
 x = 0 or 180
2Sin2(x) = 1
Sin2(x) = 0.5


Sin(x) = 0.707  x = 45 or x = 135
Number line
0--------------45---------------------------------------135-------------180
Sample using y
x = 30 y =
x = 90
=
+2Sin3x
-Sin(x)
0.25 – 0.5 < 0
y= 2–1>0
x = 150 y = 0.25 – 0.5 < 0
Therefore:
(-Sin(x)) x (Cos(2x)) > 0 for x ∈ [0 ; +180]
45 < x < 135
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