Question

Consider the equation 3tan²(x)-1=0

The solutions of this equation over the interval [0,π ] are____

Answer 1

The solutions of the equation 3tan²(x)-1=0 over the interval [0,π] are approximately x ≈ 0.615 and x ≈ 2.527, from the positive and negative square roots respectively.

Step-by-step explanation:

To solve the equation 3tan²(x)-1=0, we first add 1 to both sides of the equation:

3tan²(x) = 1

Next, divide both sides by 3 to isolate tan²(x):

tan²(x) = 1/3

Now, we will calculate the square root of both sides, remembering to consider the positive and negative square roots:

tan(x) = ±√(1/3)

Using a calculator or knowledge of special triangles, we find that:

tan(x) ≈ ±√(0.333...)

The positive solution for x is:

x = tan⁻¹(√(1/3))

The negative solution for x is:

x = π + tan⁻¹(-√(1/3))

We are looking for solutions over the interval [0,π], which means we consider the first and second quadrants where the tangent function is positive and negative respectively. By calculating these using the inverse tangent function, we get the two solutions:

x ≈ 0.615 (for the positive square root)

x ≈ 2.527 (for the negative square root)

These x-values are where the equation 3tan²(x)-1=0 is satisfied within the interval [0,π].