Determine the roots of the function f(x)=-2cos(3x)-1. Can you please show all the work so it is easier to understand, thank you!!
set 2cos(3x) - 1 = 0
2cos(3x) = 1
cos 3x = 1/2
I will use degrees, we can always switch to radians later
3x = 60° or 3x = 300° by the CAST rule
x = 20° or x = 100°
the period of cos 3x = 360/3 or 120°
so by adding/subtracting 120° to any of our answers we can produce as many new answers as we want,
so for 0 ≤ x ≤ 360°
x = 20°, 140°,100° , 220° , 260°, 340°
or in radians
x = π/9, 7π/9, 5π/9, 11π/9 , 13π/9, 17π/9
Ok so how did you get x to equal 20, 140,100........
And I also don't get where 3x=60 or 100 came from, can you please explain these parts by showing the math? Thank you
I knew, or my calculator told me, that
cos 60° = 1/2 or cos 300° = 1/2
so 3x had to be 60° or 3x had to be 300°
well if 3x = 60 , what is x? (divide both sides by 3)
same for 3x = 300
x = 100
(on your calc.
enter
2nd
cos
.5
=
you should get 60 )
Srry I keep asking questions, but I noticed that you left out the negative sign on the 2 when you set the equation to zero, was that a mistake or were you suppose to do it like that?
To determine the roots of the function f(x) = -2cos(3x) - 1, we need to find the values of x that make f(x) equal to zero (or f(x) = 0). In other words, we need to find the x-values where the graph of the function intersects the x-axis.
Let's start by setting f(x) equal to zero and solving for x:
-2cos(3x) - 1 = 0
To solve this equation, we'll first add 1 to both sides:
-2cos(3x) = 1
Next, divide both sides by -2:
cos(3x) = -1/2
Now, we need to find the angles whose cosine value is -1/2. Typically, we find these angles using the unit circle or the special angles in trigonometry. The two angles that have a cosine value of -1/2 are π/3 and 5π/3.
So, we can write two equations corresponding to these angles:
3x = π/3 (equation 1)
3x = 5π/3 (equation 2)
To solve equation 1, we divide both sides by 3:
x = (π/3) / 3
This simplifies to:
x = π/9
To solve equation 2, we divide both sides by 3:
x = (5π/3) / 3
This simplifies to:
x = 5π/9
Therefore, the roots of the function f(x) = -2cos(3x) - 1 are x = π/9 and x = 5π/9.
I hope this explanation helps you understand how to determine the roots of the given function.