Final answer:
To solve the equation 7 - 5 tan(x + 3) = -3, isolate the tan(x + 3) term and find its inverse tangent. Evaluate the inverse tangent using a calculator and subtract 3 to find x. Don't forget to consider the given domain.
Step-by-step explanation:
To solve the equation 7 - 5 tan(x + 3) = -3, we can start by isolating the tan(x + 3) term. Subtracting 7 from both sides gives -5 tan(x + 3) = -10. Dividing both sides by -5 gives tan(x + 3) = 2. To solve for x, we need to find the inverse tangent (tan^-1) of both sides. Using a calculator, we find that x + 3 = tan^-1(2).
Now, we need to solve for x. Subtracting 3 from both sides gives x = tan^-1(2) - 3. Evaluating tan^-1(2) using a calculator and subtracting 3 gives x ≈ -0.42 radians. However, since the given domain is 0 ≤ x < 2π, we need to find the reference angle for -0.42 radians within this domain.
The reference angle can be found by subtracting -0.42 from 2π, giving approximately 6.16 radians. Therefore, the solution in the given domain is x ≈ 6.16 radians.