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India
Class XI

Coterminal Angles

Lesson

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}$39π410π=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

Find the angle of smallest positive measure that is coterminal with a $489^\circ$489° angle.

Question 2

Consider an angle of $-58$58°.

  1. Find the angle of smallest positive measure that is coterminal with $-58$58°.

  2. Find the angle of smallest negative measure that is coterminal with $-58$58°.

  3. Which quadrant does $\left(-58\right)^\circ$(58)° lie in?

    quadrant $3$3

    A

    quadrant $1$1

    B

    quadrant $4$4

    C

    quadrant $2$2

    D

Question 3

State the expression in terms of $n$n, where $n$n represents any integer, that generates all angles coterminal with $\frac{\pi}{2}$π2.

  1. $\frac{\pi}{2}$π2$+$+$\editable{}$

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

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