Find all solutions of the equation: cot x sec x = 2 cot x.
A) x = 2ð/3 + nð or x = 4ð/3 + nð or 0 + nð
B) x = 2ð/3 + 2nð or x = 4ð/3 + 2nð or 0 + nð
C) x = ð/3 + 2nð or x = 5ð/3 + 2nð or 0 + nð
D) x = ð/3 + nð or x = 5ð/3 + nð or 0 + nð
cot x sec x = 2 cot x.
cotx secx - 2cot x = 0
cotx(secx - 2)
cotx = 0 or secx = 2
if cotx = 0
the cotangent is zero when the tangent is undefined, and the tangent is undefined at 90º (pi/2) and 270º (3pi/2)
or
cosx/sinx = 0
cosx = 0 , x = pi/2 or 3pi/2
if secx = 2
cosx = 1/2, x = pi/3 or 5pi/3
add n(2pi), where n is an integer, to get the general solutions.
Who the hell is teaching 2nd graders this!?
To find all solutions of the equation cot(x) sec(x) = 2 cot(x), we can follow these steps:
Step 1: Rearrange the equation to get rid of the fractions.
Multiply both sides of the equation by cos(x) to make the sec(x) term disappear.
cot(x) * sec(x) * cos(x) = 2 * cot(x) * cos(x)
cos(x) / sin(x) * cos(x) = 2 * cos(x) / sin(x)
cos^2(x) = 2 * cos(x) * sin(x)
Step 2: Rewrite the equation using trigonometric identities.
Using the identity cos^2(x) = 1 - sin^2(x), we have:
(1 - sin^2(x)) = 2 * cos(x) * sin(x)
Rearranging the equation further:
sin^2(x) + 2 * cos(x) * sin(x) - 1 = 0
Step 3: Solve the quadratic equation.
This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. In this case, it is easier to factor the quadratic equation:
(sin(x) + 1)(sin(x) - 1) = 0
Now we have two separate equations to solve:
sin(x) + 1 = 0 or sin(x) - 1 = 0
For sin(x) + 1 = 0, we have sin(x) = -1. One possible solution is x = -π/2 + nπ, where n is an integer.
For sin(x) - 1 = 0, we have sin(x) = 1. One possible solution is x = π/2 + nπ, where n is an integer.
So, the solutions to the equation cot(x) sec(x) = 2 cot(x) are:
A) x = 2π/3 + nπ or x = 4π/3 + nπ or 0 + nπ
Therefore, the correct answer is A) x = 2π/3 + nπ or x = 4π/3 + nπ or 0 + nπ.
To find the solutions of the equation cot(x)sec(x) = 2cot(x), we can first simplify the expression.
We know that cot(x) = cos(x)/sin(x) and sec(x) = 1/cos(x).
Substituting these values into the equation, we get:
(cos(x)/sin(x))(1/cos(x)) = 2(cos(x)/sin(x))
Next, we can cancel out the common terms on both sides of the equation:
1/sin(x) = 2
Now, to isolate sin(x), we can take the reciprocal of both sides:
sin(x) = 1/2
From trigonometric identities, we know that sin(pi/6) = 1/2. So, one solution for sin(x) = 1/2 is x = pi/6.
However, we need to find all the solutions. Since sin(x) is positive in the first and second quadrants, we can add 2π to each solution to find additional solutions.
Therefore, the solutions for sin(x) = 1/2 are x = pi/6 + 2πn, where n is an integer.
Now, let's check the answer choices to see which one matches the solution we found:
A) x = 2π/3 + nπ or x = 4π/3 + nπ or 0 + nπ
B) x = 2π/3 + 2nπ or x = 4π/3 + 2nπ or 0 + nπ
C) x = π/3 + 2nπ or x = 5π/3 + 2nπ or 0 + nπ
D) x = π/3 + nπ or x = 5π/3 + nπ or 0 + nπ
Based on the solution we found, the answer is D) x = π/3 + nπ or x = 5π/3 + nπ or 0 + nπ.
Therefore, option D is the correct answer.