For an angle in any quadrant, the acute angle between the rotating arm and the \(x\)-axis is the basic angle (reference angle), \(\alpha\). The basic angle fixes the size of every ratio; the quadrant fixes the sign.
The bowtie shows equal basic angles measured from the \(x\)-axis — never from the \(y\)-axis. Reading the quadrants anticlockwise from the first, the letters A-S-T-C name what is positive there: All, Sine, Tangent, Cosine.
Basic angle and signs in each quadrant
| Quadrant | Positive there | Angle \(\theta\) from basic angle \(\alpha\) | Sketch |
|---|---|---|---|
| I | All of \(\sin\theta\), \(\cos\theta\), \(\tan\theta\) | \(\theta=\alpha\) | |
| II | \(\sin\theta\) (and \(\operatorname{cosec}\theta\)) | \(\theta=180^\circ-\alpha\) \(\theta=\pi-\alpha\) | |
| III | \(\tan\theta\) (and \(\cot\theta\)) | \(\theta=180^\circ+\alpha\) \(\theta=\pi+\alpha\) | |
| IV | \(\cos\theta\) (and \(\sec\theta\)) | \(\theta=360^\circ-\alpha\) \(\theta=2\pi-\alpha\) |
Negative angles and angles beyond \(360^\circ\)
| Situation | Rule | Example |
|---|---|---|
| Negative angle (clockwise turn) | \(\sin(-\theta)=-\sin\theta\) \(\cos(-\theta)=\cos\theta\) \(\tan(-\theta)=-\tan\theta\) | \(\cos(-70^\circ)=\cos70^\circ\) |
| Beyond one revolution | \(\sin(\theta\pm360^\circ k)=\sin\theta\) \(\cos(\theta\pm360^\circ k)=\cos\theta\) \(\tan(\theta\pm180^\circ k)=\tan\theta\) | \(\sin410^\circ=\sin50^\circ\) |
Bring the angle into \(0^\circ\) to \(360^\circ\) first, then read off the quadrant and the basic angle. The basic angle is always measured from the \(x\)-axis.
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