In O-Level trigonometry you used sine, cosine and tangent of angles inside triangles. Additional Mathematics measures an angle as a rotation from the positive \(x\)-axis — anticlockwise for positive angles, clockwise for negative — so an angle can be any size. It also introduces the radian and three reciprocal ratios.
Degrees and radians
| Conversion | Rule | Example |
|---|---|---|
| Degrees to radians | Multiply by \(\dfrac{\pi}{180^\circ}\). | \(72^\circ=72\times\dfrac{\pi}{180}\) \(=\dfrac{2\pi}{5}\text{ rad}\) |
| Radians to degrees | Multiply by \(\dfrac{180^\circ}{\pi}\). | \(\dfrac{3\pi}{4}=\dfrac{3}{4}\times180^\circ\) \(=135^\circ\) |
The quarter-turn markers are worth memorising: \(90^\circ=\dfrac{\pi}{2}\), \(180^\circ=\pi\), \(270^\circ=\dfrac{3\pi}{2}\), \(360^\circ=2\pi\).
The six trigonometric ratios
For any point \(P(x, y)\) on the rotating arm, with \(r=\sqrt{x^2+y^2}\) always positive, the six ratios are defined below. When \(\theta\) is acute this matches SOH-CAH-TOA: \(y\) is the opposite side, \(x\) the adjacent side and \(r\) the hypotenuse.
| Main ratio | Reciprocal ratio |
|---|---|
| \(\sin\theta=\dfrac{y}{r}\) | \(\operatorname{cosec}\theta=\dfrac{1}{\sin\theta}=\dfrac{r}{y}\) |
| \(\cos\theta=\dfrac{x}{r}\) | \(\sec\theta=\dfrac{1}{\cos\theta}=\dfrac{r}{x}\) |
| \(\tan\theta=\dfrac{y}{x}\) | \(\cot\theta=\dfrac{1}{\tan\theta}=\dfrac{x}{y}\) |
Match your calculator's angle mode to the question: degree questions carry the \(^\circ\) symbol, radian questions do not. A wrong mode gives a wrong answer even with correct working.
Each quotient requires a non-zero denominator. Thus \(\tan\theta\) and \(\sec\theta\) require \(x\ne0\), while \(\mathrm{cosec}\theta\) and \(\cot\theta=x/y\) require \(y\ne0\). The identity \(\cot\theta=1/\tan\theta\) requires both \(x\ne0\) and \(y\ne0\). When \(x=0\) and \(y\ne0\), cotangent is zero but tangent is undefined.
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