solve:
cot(2theta/3)= -sqrt3
Please help, I don't know how to start and how to set it up.
Let 'x' equal (2theta/3). If cot(x) = -sqrt3, then tan(x) = -1/sqrt3. Locating Tan(x) on the unit circle brings us pi/6. So, X=pi6 is the same as (2theta/3)=pi/6. Multiply both sides by 3. 2theta=pi/2. Multiply both sides by 1/2. Theta=pi/4. <<done
and theta is between 0< theta< 2 pi
To solve cot(2θ/3) = -√3, we need to find the values of θ that satisfy this equation. We can start by using the reciprocal identity of cotangent: cot(θ) = 1/tan(θ).
Step 1: Rewrite the equation using the reciprocal identity.
1/tan(2θ/3) = -√3
Step 2: Convert cotangent to tangent.
tan(2θ/3) = -1/√3
Step 3: Take the inverse tangent (arctan) of both sides to eliminate the tangent.
arctan(tan(2θ/3)) = arctan(-1/√3)
Step 4: Simplify the right side of the equation.
arctan(tan(2θ/3)) = -π/3
Step 5: Use the periodicity of tangent to find the general solution for 2θ/3.
2θ/3 = -π/3 + nπ, where n is an integer.
Step 6: Solve for θ.
θ = (-π/3 + nπ) * 3/2
Therefore, the general solution for θ is θ = (-π/2 + nπ) * 3/2, where n is an integer.
To solve the equation cot(2theta/3) = -√3, we need to find the values of theta that satisfy this equation. Let's break down the steps to solve this equation:
Step 1: Understand the equation:
- We are given the equation cot(2theta/3) = -√3.
Step 2: Identify the inverse of the cotangent:
- The inverse of the cotangent function is the tangent function: tan(theta).
Step 3: Take the reciprocal of both sides:
- Since cot(theta) = 1/tan(theta), we can rewrite the equation as:
1/tan(2theta/3) = -√3.
Step 4: Simplify the equation:
- Multiply both sides of the equation by tan(2theta/3):
1 = -√3 * tan(2theta/3).
Step 5: Isolate the tangent term:
- Divide both sides by -√3 to isolate tan(2theta/3):
tan(2theta/3) = -1/√3.
Step 6: Evaluate the tangent angle:
- Remember that the tangent of an angle is equal to the ratio of the sine and cosine of that angle:
tan(theta) = sin(theta)/cos(theta).
Step 7: Substitute the angle with 2theta/3:
- We can substitute 2theta/3 into tan(theta) in terms of sin(2theta/3) and cos(2theta/3):
sin(2theta/3)/cos(2theta/3) = -1/√3.
Step 8: Rationalize the denominator:
- Multiply both the numerator and denominator by √3 to rationalize the denominator:
(√3 * sin(2theta/3))/(√3 * cos(2theta/3)) = -1.
Step 9: Simplify the equation:
- The square root of 3 is a common factor in both the numerator and the denominator. So, divide both sides of the equation by √3:
sin(2theta/3)/cos(2theta/3) = -1/√3.
Step 10: Identify the trigonometric identity:
- The left-hand side of the equation represents the tangent of an angle. Therefore, we can rewrite the equation as:
tan(2theta/3) = -1/√3.
At this point, we have transformed the equation cot(2theta/3) = -√3 into tan(2theta/3) = -1/√3. Now, we can find the values of theta that satisfy this equation by solving for 2theta/3, and then solving for theta.