Coterminal Angles

Angles are formed by pairs of intersecting lines. If one line is rotated fully, the angle appears to be the same although numerically different. The following diagram illustrates this idea.

Angles that are related in this way are called coterminal angles.

In general, an angle coterminal with another angle differs from it by an integer multiple of $2\pi$2π or $360^{\circ}$360∘.  

Examples

Example 1

List some negative and positive angles that are coterminal with $115^{\circ}$115∘.

We need only add or subtract multiples of $360^{\circ}$360∘ to obtain the coterminal angles. So, we have

$...,-605^{\circ},-245^{\circ},115^{\circ},475^{\circ},835^{\circ},...$...,−605∘,−245∘,115∘,475∘,835∘,...

 

Example 2

Find the coterminal angle closest to zero for $\frac{39\pi}{4}$39π4​.

The number $\frac{39\pi}{4}$39π4​ is between $9\pi$9π and $10\pi$10π. We try subtracting $5\times2\pi$5×2π. This gives $\frac{39\pi}{4}-10\pi=-\frac{\pi}{4}$39π4​−10π=−π4​, which is indeed the closest coterminal angle to zero.

 

Example 3

In which quadrant does the angle $7491^{\circ}$7491∘ lie?

Here, the strategy will be to remove integer multiples of $360^{\circ}$360∘ until an angle between $0^{\circ}$0∘ and $360^{\circ}$360∘ is reached. By division, we see that $20\times360<7491<21\times360$20×360<7491<21×360. So, we subtract $20\times360^{\circ}$20×360∘ and obtain $291^{\circ}$291∘. This angle is greater than $270^{\circ}$270∘ and less than $360^{\circ}$360∘ and is therefore in the fourth quadrant.

 

More Worked Examples

Question 1

Question 2

Question 3