Question

Find an angle in each quadrant with a common reference angle with 112°, from 0°≤θ<360°

Answer 1

In each quadrant, there is an angle with a common reference angle with 112°. The angles are 112°, 248°, 248°, and 292° for the first, second, third, and fourth quadrants respectively.

Step-by-step explanation:

In each quadrant, there is an angle with a common reference angle with 112°. Let's find these angles:

- In the first quadrant, the angle will be 112°.

- In the second quadrant, the reference angle is obtained by subtracting 180° from 112°, which gives 68°. Therefore, the angle in the second quadrant is 180° + 68° = 248°.

- In the third quadrant, the reference angle is obtained by subtracting 180° from 112°, which gives 68°. Therefore, the angle in the third quadrant is 180° + 68° = 248°.

- In the fourth quadrant, the reference angle is obtained by subtracting 180° from 112°, which gives 68°. Therefore, the angle in the fourth quadrant is 360° - 68° = 292°.

Answer 2

112° in the first quadrant

68° in the second quadrant

292° in the third quadrant

248° in the fourth quadrant

Step-by-step explanation:

To find an angle in each quadrant with a common reference angle with 112°, we need to first determine the reference angle.

The reference angle is the acute angle between the terminal side of the angle and the x-axis. To find the reference angle, we subtract the nearest multiple of 180° from the given angle, 112°, so that the result is between 0° and 180°:

Reference angle = 112° - 180° = -68°

Since the reference angle is negative, we can find the corresponding positive angle by adding 180°:

Reference angle = -68° + 180° = 112°

Now that we have the reference angle, we can find an angle in each quadrant with that reference angle as follows:

First quadrant: To find an angle in the first quadrant with a reference angle of 112°, we simply take the reference angle itself since it is already acute and positive:

θ = 112°

Second quadrant: To find an angle in the second quadrant with a reference angle of 112°, we subtract the reference angle from 180°:

θ = 180° - 112° = 68°

Third quadrant: To find an angle in the third quadrant with a reference angle of 112°, we add the reference angle to 180°:

θ = 180° + 112° = 292°

Fourth quadrant: To find an angle in the fourth quadrant with a reference angle of 112°, we subtract the reference angle from 360°:

θ = 360° - 112° = 248°

Therefore, the angles with a common reference angle of 112° in each quadrant, from 0°≤θ<360°, are:

112° in the first quadrant

68° in the second quadrant

292° in the third quadrant

248° in the fourth quadrant