Use the unit circle to identify the angle's quadrant, find its reference angle, and apply the correct sign to a known exact trigonometric ratio. A point on the unit circle has coordinates , while .
Use the unit circle to identify the angle's quadrant, find its reference angle, and apply the correct sign to a known exact trigonometric ratio. A point on the unit circle has coordinates , while .
For an angle , first find the reference angle , the acute angle between the terminal side and the -axis. Evaluate the trigonometric ratio using , then determine its sign from the quadrant.
| Quadrant | Reference angle | Signs of ratios |
|---|---|---|
| I | , , | |
| II | , , | |
| III | , , | |
| IV | , , |
For example, find the ratios when . The angle is in quadrant II, so
Using the exact ratios for and the quadrant-II signs,
The magnitude of each ratio matches the corresponding ratio for the reference angle. Only its sign changes. For angles outside , first add or subtract multiples of to find a coterminal angle in this interval.
A common misconception is that every ratio has the same sign within a quadrant. This is incorrect: sine and cosine are determined by the - and -coordinates respectively, while tangent's sign follows from their quotient. Another mistake is using the original angle instead of the acute reference angle when recalling exact values.
This is SL 3.5: Unit-circle definitions and exact trigonometric ratios, studied in AA SL and HL. It can appear on Paper 1 or Paper 2. On Paper 1, show the quadrant and reference angle before giving exact values; unsupported decimals will not demonstrate the required method. On Paper 2, check the calculator's angle mode. Radians are assumed unless degrees are explicitly stated, and all solutions must lie in any specified finite interval.