findall exact solutions ( in terms of radians) to the equation 2cos(40)-radical 3 =0
There is no variable in your equation, so there is nothing to solve for.
You can rewrite it as
cos(40) = (sqrt3)/2
but that is not true, whether the 40 is in degrees or radians.
is that 40 meant to say 4(theta) ?
If so, I will use 4x instead of 4theta for easier typing
2cos 4x - √3 = 0
cos 4x = √3/2
from my 30-60-90 triangle, I know cos 30 = √3/2
and the cosine is positive in quadrants I and IV
so 4x = 30 degrees or 4x = 330 degrees
x = 7.5 degrees or x = 82.5 degrees
in radians
x = pi/24 or 11pi/24
the period of cos 4x is pi/2 or 90 degrees
so by adding multiples 90 or pi/2 to each answer we can generate more answers
x = 7.5, 97.5, 187.5, 277.5, ...
or x = 82.5, 172.5, 262.5, ....
general solution:
x = pi/24 + k(pi/2) or 11pi/4 + k(pi/2), where k is an integer.
To find exact solutions in terms of radians to the equation 2cos(40°) - √3 = 0, we need to isolate the cosine term and then solve for the angle.
Step 1: Rearrange the equation to isolate the cosine term:
2cos(40°) = √3
Step 2: Divide both sides of the equation by 2:
cos(40°) = √3/2
Step 3: Take the inverse cosine (or arc cosine) of both sides to find the angle:
40° = cos^(-1)(√3/2)
Step 4: Evaluate the inverse cosine using a calculator or trigonometric table. The result will be in radians.
Using a calculator, we find that cos^(-1)(√3/2) ≈ 0.5236 radians.
Therefore, one exact solution in terms of radians to the equation 2cos(40°) - √3 = 0 is approximately 0.5236 radians.