which of the following is a solution to 3tan^3x=tanx?
A.) 60 degrees
B.) 120 degrees
c.)150 degrees
D.)240 degrees
E.)300 degrees
To find the solution to the equation 3tan^3x = tanx, we first need to simplify the equation. Here's how you can do it:
Step 1: Let's substitute a variable for tanx, such as y. Rewrite the equation as follows: 3y^3 = y.
Step 2: Subtract y from both sides: 3y^3 - y = 0.
Step 3: Factor out y: y(3y^2 - 1) = 0.
Step 4: Set each factor equal to zero: y = 0 or 3y^2 - 1 = 0.
For the first factor, y = 0. So, we have one possible solution: tanx = 0.
For the second factor, solve for y: 3y^2 - 1 = 0. Add 1 to both sides to eliminate the negative term: 3y^2 = 1. Divide both sides by 3: y^2 = 1/3.
Now, we can consider the value of y. Since y represents tanx, we need to determine the values of x corresponding to the values of y. Keep in mind that tanx repeats itself every 180 degrees.
To find the value of y for tanx, take the square root of both sides: y = ±√(1/3).
Now, let's find the corresponding x-values for y. Calculate the inverse tangent (tan^(-1)) of √(1/3) and -√(1/3) to determine the angles.
Using a calculator or reference, we find that tan^(-1) (√(1/3)) is approximately 30 degrees, which means tan(30 degrees) = √(1/3).
Similarly, tan^(-1) (-√(1/3)) is approximately -30 degrees, which means tan(-30 degrees) = -√(1/3).
Since tanx repeats every 180 degrees, we can add or subtract multiples of 180 degrees to these angles to find additional solutions.
From the given answer choices:
A.) 60 degrees is equivalent to 60 + 180 = 240 degrees.
B.) 120 degrees is equivalent to 120 + 180 = 300 degrees.
C.) 150 degrees is equivalent to 150 + 180 = 330 degrees.
D.) 240 degrees remains the same.
E.) 300 degrees remains the same.
Now, we have the following potential solutions for x in degrees: -30, 30, 240, 300, and 330.
Comparing these possible solutions to the answer choices, we see that:
A.) 60 degrees is equivalent to 240 degrees, which is one of the solutions we found.
B.) 120 degrees is equivalent to 300 degrees, which is also one of the solutions we found.
C.) 150 degrees is not one of the solutions we found.
D.) 240 degrees is one of the solutions we found.
E.) 300 degrees is one of the solutions we found.
Therefore, the solutions to the equation 3tan^3x = tanx are 60 degrees and 300 degrees (choices A and E).
well, you have
tan^2 x = 1/3
see previous posting for solving it.