directions: draw the given angle in standard position on the graph - make sure to include any rotations! then find the rest of the information
1 160° quadrant
positive coterminal angle
negative coterminal angle
reference angle
2. 280° quadrant
positive coterminal angle
negative coterminal angle
reference angle
3. 400° quadrant
positive coterminal angle
negative coterminal angle
reference angle
4 185° quadrant
positive coterminal angle
negative coterminal angle
reference angle
5 810° quadrant
positive coterminal angle
negative coterminal angle
reference angle

Question

directions: draw the given angle in standard position on the graph - make sure to include any rotations! then find the rest of the information
1 160° 		quadrant 				
		positive coterminal angle 			
		negative coterminal angle 			
		reference angle 			
2. 280° 		quadrant 				
		positive coterminal angle 			
		negative coterminal angle 			
		reference angle 			
3. 400° 		quadrant 				
		positive coterminal angle 			
		negative coterminal angle 			
		reference angle 			
4 185° 		quadrant 				
		positive coterminal angle 			
		negative coterminal angle 			
		reference angle 			
5 810° 		quadrant 				
		positive coterminal angle 			
		negative coterminal angle 			
		reference angle

Accepted Answer

### Problem 1: $160^\circ$
# Explanation:
## Step 1: Positive Coterminal Angle
To find a positive coterminal angle, we add $360^\circ$ to the given angle.
$160^\circ + 360^\circ = 520^\circ$

## Step 2: Negative Coterminal Angle
To find a negative coterminal angle, we subtract $360^\circ$ from the given angle.
$160^\circ - 360^\circ = -200^\circ$

## Step 3: Reference Angle
Since $160^\circ$ is in the second quadrant, the reference angle is $180^\circ - 160^\circ = 20^\circ$ (the original reference angle of $180^\circ$ was incorrect).

# Answer:
- Positive coterminal angle: $520^\circ$
- Negative coterminal angle: $-200^\circ$
- Reference angle: $20^\circ$

### Problem 2: $280^\circ$
# Explanation:
## Step 1: Positive Coterminal Angle
Add $360^\circ$ to $280^\circ$:
$280^\circ + 360^\circ = 640^\circ$ (the original $-80$ was incorrect, as $-80$ is negative).

## Step 2: Negative Coterminal Angle
Subtract $360^\circ$ from $280^\circ$:
$280^\circ - 360^\circ = -80^\circ$ (this part was correct).

## Step 3: Reference Angle
Since $280^\circ$ is in the fourth quadrant, the reference angle is $360^\circ - 280^\circ = 80^\circ$.

# Answer:
- Positive coterminal angle: $640^\circ$
- Negative coterminal angle: $-80^\circ$
- Reference angle: $80^\circ$

### Problem 3: $400^\circ$
# Explanation:
## Step 1: Simplify the Angle
First, find the coterminal angle within $0^\circ - 360^\circ$:
$400^\circ - 360^\circ = 40^\circ$ (so $40^\circ$ is the positive coterminal angle, which was correct).

## Step 2: Negative Coterminal Angle
Subtract $360^\circ$ from $40^\circ$ (or from $400^\circ$):
$40^\circ - 360^\circ = -320^\circ$ (the original $-320$ was correct).

## Step 3: Reference Angle
Since $40^\circ$ is in the first quadrant, the reference angle is $40^\circ$ (correct).

# Answer:
- Positive coterminal angle: $40^\circ$ (or $400^\circ + 360^\circ = 760^\circ$, but $40^\circ$ is simpler)
- Negative coterminal angle: $-320^\circ$
- Reference angle: $40^\circ$

### Problem 4: $185^\circ$
# Explanation:
## Step 1: Positive Coterminal Angle
Add $360^\circ$ to $185^\circ$:
$185^\circ + 360^\circ = 545^\circ$

## Step 2: Negative Coterminal Angle
Subtract $360^\circ$ from $185^\circ$:
$185^\circ - 360^\circ = -175^\circ$

## Step 3: Reference Angle
Since $185^\circ$ is in the third quadrant, the reference angle is $185^\circ - 180^\circ = 5^\circ$ (this part was correct).

# Answer:
- Positive coterminal angle: $545^\circ$
- Negative coterminal angle: $-175^\circ$
- Reference angle: $5^\circ$

### Problem 5: $810^\circ$
# Explanation:
## Step 1: Simplify the Angle
Find the coterminal angle within $0^\circ - 360^\circ$ by subtracting $360^\circ$ twice:
$810^\circ - 360^\circ = 450^\circ$
$450^\circ - 360^\circ = 90^\circ$

## Step 2: Quadrant
$90^\circ$ lies on the positive y - axis (between the first and second quadrants, but typically considered as the boundary).

## Step 3: Positive Coterminal Angle
Add $360^\circ$ to $810^\circ$: $810^\circ+360^\circ = 1170^\circ$ (or use the simplified $90^\circ$ and add $360^\circ$: $90^\circ + 360^\circ=450^\circ$)

## Step 4: Negative Coterminal Angle
Subtract $360^\circ$ from $810^\circ$ three times: $810^\circ-3\times360^\circ=810 - 1080=-270^\circ$ (or from $90^\circ$: $90^\circ - 360^\circ=-270^\circ$)

## Step 5: Reference Angle
For $90^\circ$ (or angles on the axes), the reference angle is typically considered as $90^\circ$ (or $0^\circ$ depending on convention, but since it’s on the y - axis, the reference angle can be $90^\circ$).

# Answer:
- Quadrant: Boundary (between I and II, or considered as the y - axis)
- Positive coterminal angle: $1170^\circ$ (or $450^\circ$, or $90^\circ$)
- Negative coterminal angle: $-270^\circ$
- Reference angle: $90^\circ$ (or $0^\circ$ depending on context)

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QUESTION IMAGE

Question

Explanation:

Problem 1: \(160^\circ\)

Step 1: Positive Coterminal Angle

Step 2: Negative Coterminal Angle

Step 3: Reference Angle

Step 1: Positive Coterminal Angle

Step 2: Negative Coterminal Angle

Step 3: Reference Angle

Step 1: Simplify the Angle

Step 2: Negative Coterminal Angle

Step 3: Reference Angle

Answer:

Problem 2: \(280^\circ\)