solutions to the equation tan^2x - 3 = 0 ( there can be more than one)
2pi/3
pi
-pi/3
5pi/3
To find the solutions to the equation tan^2(x) - 3 = 0, we can use the following steps:
Step 1: Add 3 to both sides of the equation: tan^2(x) = 3.
Step 2: Take the square root of both sides: tan(x) = ±√3.
Step 3: Solve for x by taking the inverse tangent (arctan) of both sides:
- x = arctan(√3) and x = arctan(-√3).
Step 4: Since the tangent function has a period of π, we need to find the general solutions by adding or subtracting integer multiples of π.
Hence, the general solutions for x are:
- x = arctan(√3) + nπ, where n is an integer.
- x = arctan(-√3) + nπ, where n is an integer.
Now let's find numerical values for x by substituting values for n.
For n = 0:
- x = arctan(√3) + 0π = arctan(√3).
For n = 1:
- x = arctan(√3) + 1π.
For n = -1:
- x = arctan(√3) - 1π.
Using the same approach, we can find solutions for x = arctan(-√3) + nπ (where n is an integer).
Therefore, the solutions for the equation tan^2(x) - 3 = 0 are:
- x = arctan(√3),
- x = arctan(√3) + π,
- x = arctan(√3) - π,
- x = arctan(-√3),
- x = arctan(-√3) + π,
- x = arctan(-√3) - π.
Converting these solutions into decimal approximate values, we have:
- x ≈ 1.0472 (radians),
- x ≈ 4.1888 (radians) ≈ 2π/3 (radians),
- x ≈ -2.0944 (radians) ≈ -π/3 (radians),
- x ≈ -1.0472 (radians) ≈ -π/6 (radians),
- x ≈ 2.0944 (radians) ≈ 2π/3 (radians),
- x ≈ -4.1888 (radians) ≈ -5π/3 (radians).
Hence, the solutions to the equation tan^2(x) - 3 = 0 are 2π/3, π, -π/3, and 5π/3.
There are in fact an infinite number of solutions
I would try each of the choices
I have my calculator set on degrees so 2pi = 360
tan^2 (360/3 ) = 3 Yes that works
tan 180 does not work
tan -60 does not work
tan 300 = nope
so only A