Determine which of the following pairs of angles are co-terminal. i) 210°, -150° ii) 360°, -30° iii) -180°, 540° iv) -405°, 675° vi) 860°, 580° vi) 900°, -900°

Steps to Solve

1. Identify Co-tangent Condition

To determine if two angles are co-terminal, check if their difference is a multiple of 360 degrees.

2. Calculate Differences

For each pair:

• For 210° and -150°: $$ 210 - (-150) = 210 + 150 = 360 $$
• For 360° and -30°: $$ 360 - (-30) = 360 + 30 = 390 $$
• For -180° and 540°: $$ -180 - 540 = -720 $$
• For -405° and 675°: $$ -405 - 675 = -1080 $$
• For 860° and 580°: $$ 860 - 580 = 280 $$
• For 900° and -900°: $$ 900 - (-900) = 900 + 900 = 1800 $$

3. Check Multiples of 360

Now, check if each computed difference is a multiple of 360:

• For $360$: Yes ($360 \div 360 = 1$)
• For $390$: No ($390 \div 360 \approx 1.083$)
• For $-720$: Yes ($-720 \div 360 = -2$)
• For $-1080$: Yes ($-1080 \div 360 = -3$)
• For $280$: No ($280 \div 360 \approx 0.778$)
• For $1800$: Yes ($1800 \div 360 = 5$)

Thus, the pairs that are co-terminal are:

1. ( (210°, -150°) )
2. ( (-180°, 540°) )
3. ( (-405°, 675°) )
4. ( (900°, -900°) )