Angles in standard position; reference angles; coterminal angles
Angles in Standard Position
Angles in Standard Position is a topic in Trigonometric Functions in the Common Core State Standards. It is aligned to Standard F-TF.2, which requires students to use the unit circle to extend the trigonometric functions to all real numbers.
An angle in standard position has its vertex at the origin and initial side on the positive \(x\)-axis; coterminal angles share a terminal side, and the reference angle is the acute angle to the \(x\)-axis.
Theory
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive \(x\)-axis. Positive angles open counterclockwise; negative ones clockwise.
- Coterminal angles share the same terminal side; add or subtract full turns (\(360^\circ\) or \(2\pi\)) to find them.
- Reference angle: the acute angle between the terminal side and the \(x\)-axis. It gives the size of the trig values; the quadrant gives the sign.
- Quadrant: which of the four regions the terminal side falls in.
Coterminal and reference angles by quadrant:
How to work with standard-position angles
- Find a coterminal angle in \([0^\circ,360^\circ)\) by adding/subtracting \(360^\circ\).
- Identify the quadrant of the terminal side.
- Compute the reference angle with the quadrant rule.
- Use it to evaluate trig functions, attaching the sign for that quadrant.
\(150^\circ\) is in Quadrant II, so the reference angle is \(180^\circ-150^\circ\).
| \(180^\circ-150^\circ\) | \(=\) | \(30^\circ\) |
Subtract a full turn \(360^\circ\).
| \(400^\circ-360^\circ\) | \(=\) | \(40^\circ\) |
Add \(360^\circ\).
| \(-70^\circ+360^\circ\) | \(=\) | \(290^\circ\) |
\(290^\circ\) is in Quadrant IV.
\(\dfrac{5\pi}{6}\) is in Quadrant II; subtract from \(\pi\).
| \(\pi-\dfrac{5\pi}{6}\) | \(=\) | \(\dfrac{\pi}{6}\) |
Common pitfalls
Frequently asked questions
What does standard position mean?
The angle's vertex is at the origin and its initial side lies on the positive \(x\)-axis, with positive angles measured counterclockwise.
What are coterminal angles?
Angles that share the same terminal side. You find them by adding or subtracting full turns of \(360^\circ\) or \(2\pi\).
What is a reference angle?
The acute angle between the terminal side and the \(x\)-axis. Trig values at an angle equal those at its reference angle, up to sign.
How do you find the quadrant of an angle?
Reduce to a coterminal angle in \([0^\circ,360^\circ)\): 0-90 is QI, 90-180 QII, 180-270 QIII, 270-360 QIV.