Question

Part A:

Determine 3 angles between 0° and 360° which have the same reference angle as a rotation angle of 72°. Name at least 1 angle greater than 360° with this same reference angle.
Part B:
What if the given rotation angle in quadrant 1 is n°. Determine 3 angles between 0° and 360° which have the same reference angle as a rotation angle of n°. Explain how you determined these angle measures.

Answer 1

To find angles with the same reference angle as 72° between 0° and 360°, consider each quadrant. The other angles are 108°, 252°, and 288°. To find an angle greater than 360° with the same reference angle, add or subtract multiples of 360°, such as 432°.

Step-by-step explanation:

To determine angles between 0° and 360° that have the same reference angle as a rotation angle of 72°, we need to consider each quadrant in the coordinate plane.

In the first quadrant, the reference angle is the same as the rotation angle. So, one angle in the first quadrant is 72°.

In the second, third, and fourth quadrants, the reference angle is the complementary angle to the rotation angle in the first quadrant. Therefore, the other angles with the same reference angle of 72° are 180° - 72° = 108° in the second quadrant, 180° + 72° = 252° in the third quadrant, and 360° - 72° = 288° in the fourth quadrant.

As for an angle greater than 360° with the same reference angle, we can add or subtract multiples of 360°. So, an angle greater than 360° with a reference angle of 72° could be 360° + 72° = 432°.

Answer 2

Part A:
A reference angle is defined as the acute angle formed between a given angle and the x-axis in the first quadrant. For a rotation angle of 72°, the reference angle is 72°.

72° (same as the given rotation angle)
72° + 360° = 432°
72° + 2 * 360° = 792°
An angle greater than 360° with the same reference angle as 72° would be 72° + 360° * k, where k is any integer greater than 1. For example: 72° + 2 * 360° = 792°.

Part B:
Given a rotation angle of n° in quadrant 1, the reference angle will be n°. To find 3 angles between 0° and 360° with the same reference angle as n°, we can use the formula n° + 360° * k, where k is an integer between 0 and 2 (excluding 0).

For example, given a rotation angle of 120°, the angles with the same reference angle would be:

120° (same as the given rotation angle)
120° + 360° = 480°
120° + 2 * 360° = 840°
Therefore, we can determine the angles with the same reference angle as a given rotation angle of n° by adding multiples of 360° to the given angle