Find all solutions for the equation 3cos(x)=cot(x).
When do I use +2kpi? Is it only when I'm suppose to "find all solutions"?
I got sin(x)=1/3. Is that correct?
3 cos x = cos x / sin x
trick is that if cos x = 0, that is a solution so
x = pi/2 or x = 3 pi/2 are solutions
as well as when sin x = 1/3
sin^-1(1/3) = 19.5 degrees or .34 radians
or 180 - 19.5 = 160.5 degrees or 2.8 radians
To find the solutions for the equation 3cos(x) = cot(x), we need to rearrange the equation in terms of trigonometric functions that we can easily solve.
First, let's recall the definitions of the cosine (cos) and cotangent (cot) functions:
cos(x) = 1/sin(x)
cot(x) = cos(x)/sin(x)
Substituting these definitions into the equation, we have:
3(1/sin(x)) = cos(x)/sin(x)
Now, we can eliminate the fraction by multiplying both sides by sin(x):
3 = cos(x)
This tells us that the cosine of x is equal to 3.
To find the solutions, we can use the inverse cosine function (also known as arccos or cos^(-1)). Taking the inverse cosine of both sides, we get:
x = arccos(3)
However, there is an important detail to consider. The range of the inverse cosine function is from 0 to π, or from 0° to 180°. But we are looking for all solutions, which means we need to find additional solutions that may exist outside this range.
To find these additional solutions, we can use the periodicity of the cosine function. The cosine function has a period of 2π (or 360°), which means it repeats itself every 2π (or 360°) units.
To account for all possible solutions, we can use the general form:
x = arccos(3) + 2kπ
Here, k is an integer and it allows us to obtain an infinite number of solutions by adding multiples of 2π. By varying the value of k, we can find all the solutions within the desired interval or range.
So, in conclusion, the solutions for the equation 3cos(x) = cot(x) are given by:
x = arccos(3) + 2kπ, where k is an integer
Now, let's address your question about when to use +2kπ. When solving trigonometric equations, adding +2kπ (where k is an integer) is necessary when looking for all solutions due to the periodicity of trigonometric functions. This ensures that we capture all possible solutions within the given interval or range.
Regarding your statement that sin(x) = 1/3, it seems that there might have been an error in your calculations. Solving the equation 3cos(x) = cot(x) should lead to x = arccos(3), not sin(x) = 1/3.
I hope this explanation clarifies the process of finding solutions to the given equation. If you have any further questions, please let me know!