Solve the equation.
(tan^2)3x=3
Tan^2(3x) = 3
Apply square root on both sides of the equation
tan 3x = + sqrt(3) or -sqrt(3)= tan(π /3) or - tan(π/3)
Apply inverse tangent to both sides of equation
3x = π/3 + kπ or -π/3 + kπ where k is integer
Divide both sides by 3
x = π/9 + kπ/3 or -π/9 + kπ/3 which implies
x = ±π/9 + kπ/3
Alright, let's have some equation-solving fun! I hope you're ready for a "tan"-tastic time…
To solve the equation (tan^2)(3x) = 3, we'll need to follow a few steps. And don't worry, I've got your back!
Step 1: Start by taking the square root of both sides of the equation. You know, just to "square" things away.
√((tan^2)(3x)) = √3
Step 2: Now, remember that the square root of tan^2(3x) is just plain old tan(3x). So let's substitute that in!
tan(3x) = √3
Step 3: Finally, since we're in the mood for some total "tan"-silvania, let's take the inverse tangent (or arctan) of both sides to isolate x.
x = arctan(√3)/3
Voila! There you have it. The solution to the equation (tan^2)(3x) = 3 is x = arctan(√3)/3. Hope this answer tickled your mathematical funny bone!
To solve the equation (tan^2)3x=3, we can start by taking the square root of both sides:
√((tan^2)3x) = √3
Simplifying the left side, we have:
tan(3x) = √3
Next, we can take the inverse tangent (arctan) of both sides to isolate the angle 3x:
3x = arctan(√3)
Finally, dividing both sides by 3 gives us the solution for x:
x = (1/3) * arctan(√3)
Therefore, the solution to the equation (tan^2)3x=3 is x = (1/3) * arctan(√3).
To solve the equation (tan^2)3x = 3, we can use the trigonometric identity for tangent, which states that (tan^2)x = sec^2x - 1. Let's rewrite the equation using this identity:
(sec^2)3x - 1 = 3
Next, we can simplify the equation by substituting sec^2x = 1 + tan^2x:
(1 + tan^2(3x)) - 1 = 3
Now, we have:
tan^2(3x) = 3
To isolate tan(3x), we can take the square root of both sides:
√(tan^2(3x)) = √3
tan(3x) = ±√3
Remember that tangent is positive in quadrants I and III. Therefore, we have two cases to consider:
Case 1: tan(3x) = √3
In this case, we need to find the angle whose tangent is √3. We know that the tangent of a 60-degree angle is √3, so we can write:
3x = 60° + n * 180°
where n is an integer representing the number of complete 180-degree rotations.
Case 2: tan(3x) = -√3
In this case, we need to find the angle whose tangent is -√3. We know that the tangent of a 120-degree angle is -√3, so we can write:
3x = 120° + n * 180°
where n is an integer representing the number of complete 180-degree rotations.
By solving these equations for x, we can find the values of x that satisfy the original equation.