Solve the equation for exact solutions over the interval [0, 2pi].
4 cotx+2=-2
you want cot x = -1
cot pi/4 = 1
so you want an angle in QII or QIV
cot 3pi/4 = cot 7pi/4 = -1
To solve the equation 4 cot(x) + 2 = -2 over the interval [0, 2π], we can follow these steps:
Step 1: Subtract 2 from both sides of the equation:
4 cot(x) = -4
Step 2: Divide both sides by 4:
cot(x) = -1
Step 3: Rewrite the equation in terms of sine and cosine:
cot(x) = cos(x)/sin(x) = -1
Step 4: Rewrite -1 as a ratio of sine and cosine:
cos(x)/sin(x) = -1
Step 5: Cross multiply:
cos(x) = -sin(x)
Step 6: Divide both sides by sin(x):
cos(x)/sin(x) = -1/sin(x)
Step 7: Recognize the identity tan(x) = sin(x)/cos(x):
1/tan(x) = -1/sin(x)
Step 8: Invert both sides of the equation:
tan(x) = -sin(x)
Step 9: Apply the tangent addition formula:
tan(x) = tan(x + π)
Step 10: Solve for x + π = 0:
x + π = 0
Step 11: Subtract π from both sides:
x = -π
So, the exact solution for x over the interval [0, 2π] is x = -π.
To solve the equation 4 cot(x) + 2 = -2, we need to isolate the cot(x) term and find the angle values for which this equation holds true over the interval [0, 2pi].
Step 1: Subtract 2 from both sides of the equation:
4 cot(x) = -4
Step 2: Divide both sides of the equation by 4:
cot(x) = -1
Now we need to find the values of x for which the cot(x) is equal to -1.
The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle. It can be calculated using the formula cot(x) = cos(x) / sin(x).
Step 3: Rewrite the equation using the cosine and sine functions:
cos(x) / sin(x) = -1
Step 4: Multiply both sides of the equation by sin(x):
cos(x) = -sin(x)
Step 5: Square both sides of the equation to eliminate the cosine term:
cos^2(x) = sin^2(x)
Using the Pythagorean Identity cos^2(x) + sin^2(x) = 1, we substitute cos^2(x) = 1 - sin^2(x) into the equation:
1 - sin^2(x) = sin^2(x)
Step 6: Simplify the equation:
1 = 2sin^2(x)
Step 7: Divide both sides of the equation by 2:
1/2 = sin^2(x)
Taking the square root of both sides, we get:
sin(x) = ±√(1/2)
Step 8: Solve for x using sin(x) = ±√(1/2):
Case 1: sin(x) = √(1/2)
To find the angle whose sine is √(1/2), we consider the unit circle and the values of the sine function.
sin(π/4) = √(1/2)
sin(3π/4) = √(1/2)
Since we are looking for x values over the interval [0, 2pi], the solutions for this case are x = π/4 and x = 3π/4.
Case 2: sin(x) = -√(1/2)
Similarly, we can find the angle whose sine is -√(1/2) on the unit circle.
sin(5π/4) = -√(1/2)
sin(7π/4) = -√(1/2)
Again, considering the interval [0, 2pi], the solutions for this case are x = 5π/4 and x = 7π/4.
Therefore, the exact solutions for the equation 4 cot(x) + 2 = -2 over the interval [0, 2pi] are x = π/4, x = 3π/4, x = 5π/4, and x = 7π/4.