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Locate extended angles and generate coterminal angles

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When reducing an extended angle, which full-turn decision should be made first?

Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, No calculatorBuild a worksheet

Coterminal angles and terminal position

Reduce an angle by complete turns until its representative lies in the principal interval. Use the reduced angle for the quadrant or axis, then add or subtract complete turns to make other coterminal angles

\[\theta_{\rm cot}=\theta_0+kT\]
  1. Use one full turn Choose one full turn in the angle unit being used \(\displaystyle T=360^\circ\quad\text{or}\quad T=2\pi\text{ rad}\)
  2. Reduce to the principal interval Choose the representative in the interval from zero inclusive to one full turn exclusive \(\displaystyle 0^{\circ}\le\theta_0<360^{\circ},\quad 0\le\theta_0<2\pi\)
  3. Use the sign for direction A positive rotation is anticlockwise and a negative rotation is clockwise \(\displaystyle +360^{\circ}\text{ is anticlockwise},\quad -360^{\circ}\text{ is clockwise}\)
  4. Add full turns for coterminal angles Angles with the same terminal position differ by an integer number of full turns \(\displaystyle \theta_{\rm cot}=\theta_0+kT,\quad k\in\mathbb{Z}\)

Worked example

Reduce the angle below to the principal interval, identify its terminal position and give one positive coterminal angle

\[\theta=1050^{\circ}\]
  1. \(\displaystyle T=360^{\circ},\quad 1050^{\circ}=2\left(360^{\circ}\right)+330^{\circ}\)Use 360 degrees for each complete turn and leave the remainder in the principal interval
  2. \(\displaystyle \theta_0=330^{\circ}\)The reduced angle is the remainder after two complete turns are removed
  3. \(\displaystyle 270^{\circ}<330^{\circ}<360^{\circ}\quad\Rightarrow\quad\text{fourth quadrant}\)An angle strictly between 270 degrees and 360 degrees has its terminal side in the fourth quadrant
  4. \(\displaystyle 330^{\circ}+360^{\circ}=690^{\circ}\)Adding one full turn gives a positive coterminal angle

Common mistakes

Using a half-turn instead of a full turn
✕\(\displaystyle 450^{\circ}=90^{\circ}+180^{\circ}=270^{\circ}\)✓\(\displaystyle 450^{\circ}=90^{\circ}+360^{\circ}\)
Coterminal angles differ by complete turns, so the amount added or subtracted must be one full turn rather than a half-turn
Reading position before reduction
✕\(\displaystyle 720^{\circ}\text{ lies in the second quadrant}\)✓\(\displaystyle 720^{\circ}\to0^{\circ}\text{, the positive horizontal axis}\)
Reduce an extended angle to the principal interval before comparing it with quadrant or axis boundaries
Reversing negative rotation direction
✕\(\displaystyle -90^{\circ}\text{ is anticlockwise}\)✓\(\displaystyle -90^{\circ}\text{ is clockwise}\)
The sign convention makes positive rotations anticlockwise and negative rotations clockwise

Difficulty levels

introReduce one positive degree or π-radian angle within two revolutions
standardHandle a negative rotation and produce one requested coterminal angle
challengeHandle several revolutions, mixed output signs or a rational-radian position
examChoose a terminal position or direction using full-turn and half-turn distractors

Questions about this topic

What are coterminal angles?+

Coterminal angles have the same terminal side and differ by an integer number of full turns

\[\begin{gathered}\theta_{\rm cot}=\theta_0+kT,\quad k\in\mathbb Z\\T=360^\circ\quad\text{or}\quad T=2\pi\text{ rad}\end{gathered}\]
Why is 360 degrees excluded from the principal interval?+

The terminal position at 360 degrees is the same as at 0 degrees. Using 0 inclusive and 360 exclusive gives one representative for that position

Why does a negative angle turn clockwise?+

By convention, anticlockwise rotations are positive and clockwise rotations are negative

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 9: Circular functions

In this chapter

Convert angle measure and use the radian arc relationLocate extended angles and generate coterminal anglesDeduce exact trigonometric ratios from one ratioUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsLocate and bisect a circular-equation rootUse local polynomial approximations for circular functions