Question

Tanx=-2

solve the equation giving the answer in the interval 0°≤ x ≤360°

Answer 1

Approximately 116.57° and 296.57°.

Explanation:

To solve the equation tan(x) = -2 in the interval 0° ≤ x ≤ 360°, we can follow these steps:

Find the reference angle:

Use the inverse tangent function (arctan) to find the reference angle whose tangent is 2:

arctan(2) ≈ 63.43°

Determine the principal solution:

The principal solution lies in the second quadrant, where tangent is negative.

Subtract the reference angle from 180° to find the principal solution:

Principal solution: 180° - 63.43° ≈ 116.57°

Find the general solutions:

Since the tangent function has a periodicity of 180°, we can add or subtract multiples of 180° to the principal solution to find the general solutions.

In the second quadrant:

Second solution: Principal solution + 180° ≈ 116.57° + 180° ≈ 296.57°

Therefore, the general solutions in the interval 0° ≤ x ≤ 360° are approximately 116.57° and 296.57°.

Note: The tangent function has other solutions outside the given interval, but we are considering solutions only within the range of 0° to 360°.

Answer 2

0° ≤ x ≤ 360° are x = 243.43° and x = 296.57°.

Explanation:

To solve the equation tan(x) = -2 in the interval 0° ≤ x ≤ 360°, we can use the inverse tangent function or arctan. 1. Take the inverse tangent of both sides of the equation: arctan(tan(x)) = arctan(-2). 2. The inverse tangent of tan(x) is simply x, so we have x = arctan(-2). Now, let's find the value of arctan(-2): 3. Arctan(-2) is an angle whose tangent is -2. We can use a calculator to find the approximate value, or we can use reference angles and special triangles. - Using a calculator, we find that arctan(-2) is approximately -63.43°. - Alternatively, we can consider the reference angle. Since tan is negative in the second and fourth quadrants, we know that the reference angle is arctan(2), which is approximately 63.43°. 4. To find the solutions in the given interval, we need to consider the quadrants where the tangent is negative (-2 is negative). - In the second quadrant, angles have a reference angle of 63.43°. So the solution in this quadrant is 180° + 63.43° = 243.43°. - In the fourth quadrant, angles have a reference angle of 63.43°. So the solution in this quadrant is 360° - 63.43° = 296.57°. Therefore, the solutions to the equation tan(x) = -2 in the interval 0° ≤ x ≤ 360° are x = 243.43° and x = 296.57°.