Find the value of cos2x=3/4.Find the two values of x.
0<x>pi(180)
see your previous post. You should be able to finish it off.
Check your answers to make sure they fit the original equation.
Can you show how to do with inverse of cos?
Picking up from oobleck's
cosx = ±√(7/8) = ±0.935
depends on your calculator.
on mine (SHARP) I do the following.
2nd cos
.935
=
to get 20.77...
I have it said on degrees, (the DRG) key, so one answer is 20.77°
Of course using your CAST rule, you know that the cosine is positive
in quads I and IV, so another answer is 360 - 20.77 ° or 339.23°
BUT, your domain was 0 < x < 180°, so we reject that answer
Going back we see that cosx = -.935
so x must be in quads II and III, but we only need quad II,
so x = 180 - 20.77 ° = 159.23
if you want your answer in radians, set the DRG to RAD
repeat the same steps, but use 2π and π instead of 360° and 180°
To find the values of x for which cos(2x) is equal to 3/4, we can first find the values of x for which cos(2x) is equal to the positive square root of 3/4, which is √3/2.
We know that the cosine function is positive in the first and fourth quadrants. The cosine function has a period of 2π, so we can find the values of x in the interval 0 < x < π/2 and then add π to those values to find the values in the interval π < x < 2π.
Step 1: Solve for x in the first quadrant:
cos(2x) = √3/2
We can use the inverse cosine function to solve for x:
2x = arccos(√3/2)
Since arccos(√3/2) = π/6, we have:
2x = π/6
Dividing both sides by 2, we get:
x = π/12
Step 2: Add π to x to find the value in the second quadrant:
x + π = π/12 + π
x + π = 13π/12
Step 3: Solve for x in the fourth quadrant:
cos(2x) = √3/2
Using the same logic as before, we have:
2x = -arccos(√3/2) (taking the negative angle since we are in the fourth quadrant)
2x = -π/6
x = -π/12
Step 4: Add π to x to find the value in the third quadrant:
x + π = -π/12 + π
x + π = 11π/12
So, the two values of x that satisfy cos(2x) = 3/4 and 0 < x < π are:
1) x = π/12
2) x = 13π/12
Note: It is important to consider the whole unit circle to find all possible solutions within the given interval.