3tan 1/3 theta=8 in the interval from 0 to 2pi. Solve the equation.
3tan(θ/3) = 8
tan(θ/3) = 8/3
Now, tan(1.212) = 8/3, so use that as your reference angle. tanθ is positive is QI and QIII, so
θ/3 = 1.212 or π+1.212=4.353
so, θ=3.636 or 13.061
The only solution in [0,2π) is 3.636
thank you
To solve the equation 3tan(1/3θ) = 8 in the interval from 0 to 2π, we can follow these steps:
Step 1: Rewrite the equation
tan(1/3θ) = 8/3
Step 2: Take the inverse tangent of both sides to isolate θ
1/3θ = arctan(8/3)
Step 3: Multiply both sides by 3 to solve for θ
θ = 3 * arctan(8/3)
Step 4: Use a calculator to find the value of arctan(8/3). This gives approximately 1.2762 radians.
Step 5: Multiply the result from step 4 by 3 to get the values of θ within the specified interval:
θ = 3 * 1.2762
θ ≈ 3.8286
Therefore, the solution to the equation 3tan(1/3θ) = 8 in the interval from 0 to 2π is θ ≈ 3.8286.
To solve the equation 3tan(1/3θ) = 8, we need to isolate θ. Here's how you can solve it step by step:
Step 1: Start with the given equation: 3tan(1/3θ) = 8.
Step 2: Divide both sides of the equation by 3: tan(1/3θ) = 8/3.
Step 3: Take the inverse tangent (arctan) of both sides to cancel out the tangent function: arctan(tan(1/3θ)) = arctan(8/3).
Step 4: Use the property that arctan(tan(x)) = x + nπ, where n is an integer, to simplify the left side of the equation: 1/3θ = arctan(8/3) + nπ.
Step 5: To eliminate the 1/3 factor in front of θ, multiply both sides by 3: θ = 3(arctan(8/3) + nπ).
Step 6: Since we are looking for solutions in the interval from 0 to 2π, we need to find all the values of n that satisfy this condition.
Step 7: Plug in n = 0 into the equation θ = 3(arctan(8/3) + nπ): θ = 3(arctan(8/3) + 0π) = 3arctan(8/3).
Step 8: Calculate the value of arctan(8/3) using a calculator. This will give you the numerical value of the first solution for θ in the interval.
Step 9: Repeat steps 7 and 8 for other values of n (1, 2, -1, -2, etc.), and use a calculator to determine the corresponding values of θ.
Step 10: Keep track of all the solutions for θ that fall within the interval from 0 to 2π.
After following these steps, you should be able to find all the solutions to the given equation in the specified interval.