Two angles with the same initial and terminal sides but possibly different rotations are called angles. increasing or decreasing the degree measure of an angle in standard position by an integer multiple of results in such an angle. increasing or decreasing the radian measure of an angle in standard position by an integer multiple of results in such an angle.

Two angles with the same initial and terminal sides but possibly different rotations are called Coterminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results in such an angle. Increasing or decreasing the radian measure of an angle in standard position by an integer multiple of 2π  results in such an angle.

Step-by-step explanation:

Consider the provided information.

Coterminal angles are angles that share the same sides of the initial and terminal. Depending on whether the given angle is in degrees or radians, calculating coterminal angles is as simple as adding or subtracting 360° or 2π to each angle. An angle of θ° is coterminal with angles of θ±360°k, where k is an integer.

Now fill the blanks as shown:

Two angles with the same initial and terminal sides but possibly different rotations are called Coterminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results in such an angle. Increasing or decreasing the radian measure of an angle in standard position by an integer multiple of 2π  results in such an angle.

Two angles with the same initial and terminal sides but possibly different rotations are called Coterminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results in such an angle. Increasing or decreasing the radian measure of an angle in standard position by an integer multiple of 2π  results in such an angle.

Step-by-step explanation:

Two angles in the same position are called corresponding angles. They have the same measure if the original two lines are parallel.

These are two questions and two answers.

Question 1. For all x in the domain, the function is equivalent to:

Explanation:

First note the the domain of the function is all the real values except those for which the denominator is zero, this is x³ - x = 0.

And x³ - x = 0 ⇒ x (x² - 1) = 0 ⇒ x (x - 1) (x + 1)  = 0

⇒ x = 0, x = 1, x = - 1.

∴ The domain is all the real values such that x ≠0, x ≠ - 1, and x ≠ 1.

Now, you can simplify the function following these steps:

So, the answer is the option J.

Question 2. What are the real solutions to the equation |x|² + 2|x| - 3 = 0?

option F. +/- 1

Explanation:

1) Use the fact that |x|² = x²

⇒ x² + 2|x| - 3 = 0

2) Transpose terms to isolate 2|x|:

⇒2|x|  = 3 - x²

3) Square both sides:

⇒ [2|x|] ² = (3 - x² )²

⇒ 4|x|² = 9 - 6x² + x⁴

⇒ 4x² = 9 -6x² + x⁴

4) Transpose terms:

⇒  x⁴ - 4x² - 6x² + 9 = 0

⇒ x⁴ - 10x² + 9 = 0

5) Change variable: x² = u

⇒ u² - 10u + 9 = 0

6) Factor:

⇒ (u - 9) (u - 1) = 0

6) Comeback to the variable x (undo the change of variable):

⇒ x² =  u ⇒

6) Verify extraneous solutions:

|x|² + 2|x| - 3 = 0

        ⇒ |3|² + 2|3| - 3 = 0

        ⇒ 9 + 6 - 3 = 0

        ⇒ 12 = 0 ⇒ FALSE ⇒ extraneous solution ⇒ discarded

        ⇒ |-3|² + 2|-3| - 3 = 0

        ⇒ 9 + 6 - 3 = 0

        ⇒ 12 = 0 ⇒ FALSE ⇒ extraneous solution ⇒ discarded

        ⇒ |1|² + 2|1| - 3 = 0

        ⇒ 1 + 2 - 3 = 0

        ⇒ 0 = 0 ⇒ TRUE ⇒ actual solution

        ⇒ |-1|² + 2|-1| - 3 = 0

        ⇒ 1 + 2 - 3 = 0

        ⇒ 0 = 0 ⇒ TRUE ⇒ actual solution

Conclusion: the solutions are +1 and -1.