Question

Solve the equations when 0 ≤ θ ≤ 2π:
a. tan⁡x = 0.05
b. tan⁡x = -50

Answer 1

To solve the given equations, we can take the inverse tangent (arctan) of both sides and add or subtract multiples of 2π to find the solutions within the given range.

Step-by-step explanation:

For part (a):

We need to solve the equation tan x = 0.05 for the values of x between 0 and 2π.

To find x, we can take the inverse tangent (arctan) of both sides of the equation:

x = arctan(0.05)

Using a calculator, we find that x ≈ 2.862 radians. However, since the given range is 0 ≤ θ ≤ 2π, we need to find the equivalent angle in this range. Adding or subtracting multiples of 2π, we can write the solution as:

x ≈ 2.862 + 2πn, where n is an integer

For part (b):

We need to solve the equation tan x = -50 for the values of x between 0 and 2π.

Using the same process as in part (a), we find that x ≈ -1.550 radians. Again, to find the solution within the given range, we add or subtract multiples of 2π:

x ≈ -1.550 + 2πn, where n is an integer