Find all solutions of the equation cot(theta)+1=0 through the interval [0,2pie).
same as when tan = -1
which is when theta = 90+45 =135 = 3 pi/4
or when theta = 360 - 45 = 315 = 7 pi/4
Why did the cot(theta) go to the therapist?
Because it needed help finding its solutions!
Now, let's find those solutions through the interval [0, 2π).
First, let's rewrite the equation cot(theta) + 1 = 0:
cot(theta) + 1 = 0
Since cot(theta) is the reciprocal of tan(theta), we can rewrite the equation in terms of tan(theta):
1/tan(theta) + 1 = 0
Now, let's solve for tan(theta):
1 + tan(theta) = 0
tan(theta) = -1
To find the solutions, we need to find the angles where the tangent equals -1.
In the interval [0, 2π), the angles where tan(theta) = -1 are π/4 and 5π/4.
So, the solutions to the equation cot(theta) + 1 = 0 through the interval [0, 2π) are theta = π/4 and theta = 5π/4.
Hope this helps you wrap your head around those solutions!
To find all solutions of the equation cot(theta) + 1 = 0 on the interval [0, 2pi), we can first solve for theta and then check if the solutions fall within the given interval.
1. Start with the equation cot(theta) + 1 = 0.
2. Subtract 1 from both sides to isolate cot(theta): cot(theta) = -1.
3. Taking the reciprocal of both sides, we get: tan(theta) = -1.
4. Recall that the tangent function is negative in the third and fourth quadrants.
5. Find the reference angle: tan(theta) = -1 is equivalent to theta = arctan(-1).
The reference angle for arctan(-1) is pi/4 (45 degrees).
6. Since the tangent function is negative, we need to add pi (180 degrees) to the reference angle to find the solutions in the third and fourth quadrants.
Therefore, the solutions in the third quadrant are: theta = pi + pi/4 = 5pi/4.
And the solutions in the fourth quadrant are: theta = 2pi - pi/4 = 7pi/4.
7. Check if these solutions fall within the given interval [0, 2pi).
Both 5pi/4 and 7pi/4 are within the interval [0, 2pi), so they are valid solutions.
Therefore, the solutions of the equation cot(theta) + 1 = 0 on the interval [0, 2pi) are theta = 5pi/4 and theta = 7pi/4.
To find all solutions of the equation cot(theta) + 1 = 0 through the interval [0, 2π), you can follow these steps:
Step 1: Rewrite the equation.
To eliminate the cotangent function, we can rewrite the equation as tan(theta) = -1.
Step 2: Determine where tan(theta) = -1.
Since the tangent function is negative in the second and fourth quadrants, and tan(theta) = -1 when theta = 3π/4 in the second quadrant and theta = 7π/4 in the fourth quadrant.
Step 3: Check the interval [0, 2π).
We need to check whether the solutions 3π/4 and 7π/4 fall into the interval [0, 2π). Let's evaluate their values:
- For theta = 3π/4, this value is between π/2 and π, which is within the interval [0, 2π).
- For theta = 7π/4, this value is between 2π and 3π/2, which is outside the interval [0, 2π).
Step 4: Finalize the solutions.
Since theta = 3π/4 is the only solution that falls within the interval [0, 2π), it is the only solution to the equation cot(theta) + 1 = 0 in the given interval.
Therefore, the solution to the equation cot(theta) + 1 = 0 through the interval [0, 2π) is theta = 3π/4.