Determine the solution of the equetion: 2cos x-1=0? 2cos x=0? and 2cos x+1=0?
2cosx - 1 = 0
2cosx = 1
cosx = 1/2
I know from my basic trig relations that cos 60° = 1/2 and by the CAST rule cos 300° = 1/2
so x = 60° or x - 300° , or in radians: π/3, 5π/3
2cosx = 0
cosx = 0
I know from looking at a cosine curve that
x = 90° or x = 270° or in radians : π/2, 3π/2
do the third one the same way I did the first one
The answer is 120 if i'm not mistaken
Since the cosine is negative in quads II or III, the answers would be 120° or 240°
To determine the solution to each of these equations, we'll need to solve for x. Let's break it down one equation at a time.
1) 2cos x - 1 = 0:
To solve this equation, we'll isolate the cosine term by moving the constant term to the other side of the equation:
2cos x = 1
Next, divide both sides of the equation by 2 to solve for cos x:
cos x = 1/2
Now, to find the solutions for x, we need to consider the values of cos θ that have a cosine ratio of 1/2. The common angles that satisfy this condition are π/3 and 5π/3 (or 60° and 300° in degrees).
Hence, the solutions for x in the equation 2cos x - 1 = 0 are:
x = π/3 + 2πn, where n is an integer
x = 5π/3 + 2πn, where n is an integer
2) 2cos x = 0:
To solve this equation, again, we'll isolate the cosine term:
cos x = 0
Now, for cos x to be equal to 0, we need to find the values of x that correspond to points where the cosine function is equal to 0. These points occur at π/2 and 3π/2 (or 90° and 270° in degrees).
Therefore, the solutions for x in the equation 2cos x = 0 are:
x = π/2 + πn, where n is an integer
x = 3π/2 + πn, where n is an integer
3) 2cos x + 1 = 0:
Following the same steps as before, we'll isolate the cosine term:
2cos x = -1
cos x = -1/2
Now, considering the values of cos x that equal -1/2, we find that these points occur at 2π/3 and 4π/3 (or 120° and 240° in degrees).
Thus, the solutions for x in the equation 2cos x + 1 = 0 are:
x = 2π/3 + 2πn, where n is an integer
x = 4π/3 + 2πn, where n is an integer
Please note that adding 2πn accounts for the periodic nature of the cosine function, as there are infinitely many solutions.