Hi! Okay so I started to attempt this math problem; 3cot^2x-1=0. However, I'm a little stuck. My teacher wants me to find the Location on The Unit Circle, Period, and General Solution. Can someone check what I have and help me with the rest?
3cot^2x-1=0
cot^2x=1/3
tan^2x=3
tanx=+ and - sqrt3
Location: pi/3,2pi/3,4pi/3 and 5pi/3
Period: 3pi?????
General Solution: (Not sure)
the period of cot(x) is pi. The amplitude and power make no difference.
So, the general solutions are
n*pi ± pi/3
for any integer n.
To solve the equation 3cot^2x - 1 = 0, let's break it down step by step:
Step 1: Rewrite the equation using the reciprocal identity for cotangent:
cot^2x = 1/3 can be rewritten as tan^2x = 3.
Step 2: Take the square root of both sides to solve for the tangent:
tanx = ±√3.
Step 3: Find the locations on the unit circle where the tangent is equal to ±√3.
The unit circle represents the values of sine and cosine for various angles. For a tangent value of ±√3, we can find the corresponding angles by looking at the values of sine and cosine on the unit circle.
The tangent is positive in the first and third quadrants, so we have two possible angles:
In the first quadrant, sinx = √3/2 and cosx = 1/2. This corresponds to an angle of π/3.
In the third quadrant, sinx = -√3/2, and cosx = -1/2. This corresponds to an angle of 4π/3.
So the locations on the unit circle where the ratio is equal to ±√3 are π/3 and 4π/3.
Step 4: Determine the period of the function.
The period of the cotangent function is π, which means that the graph repeats after each π radians or 180 degrees. However, in this case, we are dealing with the tangent function. The period of tangent is half of the cotangent function, so it is π/2.
Step 5: Find the general solution.
To find the general solution, we need to consider all possible values of x that satisfy the equation.
Since the period of tangent is π/2, we can add nπ/2, where n is an integer, to the angles we found earlier, π/3 and 4π/3.
So the general solution for x is x = π/3 + nπ/2 and x = 4π/3 + nπ/2.
In conclusion:
Location on the unit circle: π/3, 4π/3
Period: π/2
General solution: x = π/3 + nπ/2 and x = 4π/3 + nπ/2, where n is an integer.