find all solutions for
tan^2 theta -4 = 0
i got theta = arctan 2
is that correct? and if it is how would i write it as a solution?
theta = arctan2 or arctan -2
because tan^2 = 4, so tan is +/- 2
that's as exact as you can get. Just as pi or sqrt(7) is a perfectly valid number, so is arctan(2)
just be sure to note that since arctan(-2) = pi-arctan(2), the complete solution set is
k*pi+/-Arctan(2)
where Arctan(2) is the principal value.
To find all solutions for the equation tan^2(theta) - 4 = 0, you can start by rearranging the equation:
tan^2(theta) = 4
Next, take the square root of both sides:
tan(theta) = ±√4
Simplifying, you get:
tan(theta) = ±2
The tangent function is positive in the first and third quadrants, and negative in the second and fourth quadrants. So, it has two solutions:
1. When tan(theta) = 2:
- In the first quadrant, you can find theta by taking the inverse tangent (arctan) of 2. This gives theta = arctan(2).
2. When tan(theta) = -2:
- In the third quadrant, add π to the first-quadrant solution to get the second solution: theta = arctan(2) + π.
Therefore, the solutions for the equation tan^2(theta) - 4 = 0 are:
theta = arctan(2) and theta = arctan(2) + π.
To find all solutions for the equation tan^2(theta) - 4 = 0, we need to solve for theta.
First, let's rearrange the equation:
tan^2(theta) = 4
Next, take the square root of both sides:
tan(theta) = ±√4
Simplifying:
tan(theta) = ±2
The tangent function is positive in two quadrants: the first quadrant (0° to 90°) and the third quadrant (180° to 270°). In these quadrants, the values of theta that satisfy the equation are as follows:
In the first quadrant:
theta = arctan(2)
In the third quadrant:
theta = arctan(2) + π
So, your answer is correct and can be written as follows:
theta = arctan(2) + kπ
Here, k represents any integer (positive, negative, or zero) that determines all possible solutions. It allows us to generate an infinite number of solutions by adding or subtracting full rotations (2π) to the arctan(2) angle.