FINDING TWO COTERMINAL ANGLES FOR THE GIVEN ANGLE

Coterminal Angles are angles who share the same initial side and terminal sides.

Finding coterminal angles is as simple as adding or subtracting 360^º or 2π to each angle, depending on whether the given angle is in degrees or radians.

How many coterminal angles are possible ?

There are an infinite number of coterminal angles that can be found. Following this procedure, all coterminal angles can be found.

If more than one positive coterminal angle needs to be found, simply add another 360^º. This would essentially make the new angle complete two full revolutions before its terminal side comes to rest.

Find one negative angle that is coterminal to 30^º.

A negative angle moves in a clockwise direction. In this case, to find the negative coterminal angle, subtract 360^º from 30^º.

30^º - 360^º = -330^º

Find one negative angle that is coterminal to 150^º.

150^º - 360^º = -210^º

Find one negative angle that is coterminal to 415^º.

415^º - 360^º = 55^º

Although 55^º is a coterminal angle to 415^º, this is not a solution to the problem. The problem specifically asked for a negative angle, so the process needs to take place one more time.

55^º - 360^º = -305^º

Find two coterminal angles for each given angle.

Problem 1 :

-5^º

Solution :

The given angle is, θ = -5^º

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles :

Let n = 1

Positive angles :

θ + 360n

-5 + 360(1)

- 5 + 360

355^º

Negative angles :

θ - 360n

-5 - 360(1)

-5 - 360

-365^º

So, the angle is 355^º and -365^º.

Problem 2 :

5^º

Solution :

The given angle is, θ = 5^º

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 360n

5 + 360(1)

5 + 360

365^º

Negative angles :

θ - 360n

5 - 360(1)

5 - 360

-355^º

So, the angle is 365º and -355º.

Problem 3 :

45º

Solution :

The given angle is, θ = 45º

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 360n

45 + 360(1)

45 + 360

405º

Negative angles :

θ - 360n

45 - 360(1)

45 - 360

-315º

So, the angle is 405º and -315º.

Problem 4 :

-45º

Solution :

The given angle is, θ = -45º

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 360n

-45 + 360(1)

-45 + 360

315º

Negative angles :

θ - 360n

-45 - 360(1)

-45 - 360

-405º

So, the angle is 315º and -405º.

Problem 5 :

5π/2

Solution :

The given angle is, θ = 5π/2

5π/2 is reference angle.

So, 5π/2 = π/2

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

π/2 + 2π(1)

π/2 + 2π

5π/2

π/2

Negative angles :

θ - 2πn

π/2 - 2π(1)

π/2 - 2π

-3π/2

So, the angle is π/2 and -3π/2.

Problem 6 :

-π

Solution :

The given angle is, θ = -π

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

-π + 2π(1)

-π + 2π

Negative angles :

θ - 2πn

-π - 2π(1)

-π - 2π

-3π

So, the angle is π and -3π.

Problem 7 :

17π/9

Solution :

The given angle is, θ = 17π/9

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

17π/9 + 2π(1)

(17π + 18π)/9

35π/9

Negative angles :

θ - 2πn

17π/9 - 2π(1)

(17π - 18π)/9

-π/9

So, the angle is 35π/9 and -π/9.

Problem 8 :

13π/18

Solution :

The given angle is, θ = 13π/18

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

13π/18 + 2π(1)

(13π + 36π)/18

49π/18

Negative angles :

θ - 2πn

13π/18 - 2π(1)

(13π - 36π)/18

-23π/18

So, the angle is 49π/18 and -23π/18.

Problem 9 :

5π/4

Solution :

The given angle is, θ = 5π/4

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

5π/4 + 2π(1)

5π/4 + 2π

13π/4

Negative angles :

θ - 2πn

5π/4 - 2π(1)

5π/4 - 2π

-3π/4

So, the angle is 13π/4 and -3π/4.

Problem 10 :

25π/36

Solution :

The given angle is, θ = 25π/36

The formula to find the coterminal angles is, θ ± 2πn

Let us find two coterminal angles :

n = 1

Positive angles :

θ + 2πn

25π/36 + 2π(1)

(25π + 72π)/36

97π/36

Negative angles :

θ - 2πn

25π/36 - 2π(1)

(25π - 72π)/36

-47π/36

So, the angle is 97π/36 and -47π/36.

FINDING TWO COTERMINAL ANGLES FOR THE GIVEN ANGLE

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