Finding Trigonometric Ratios
Finding Trigonometric Ratios
IB AA SL: core and optional content
Core: signed coordinates and sin, cos and tan. Optional: reciprocal-ratio columns and the sec θ answer.
Locate the quadrant and form a right-angled triangle. Use signed coordinates \(x,y\) and a positive radius \(r\).
\(r=\sqrt{x^2+y^2}\qquad \sin\theta=\dfrac yr\qquad \cos\theta=\dfrac xr\qquad \tan\theta=\dfrac yx\)
\(r>0\). Tangent requires \(x\ne0\); reciprocal ratios require a nonzero denominator.
| Quadrant | \(x\) | \(y\) | Positive ratios |
|---|---|---|---|
| I | + | + | All |
| II | − | + | \(\sin\theta,\ \operatorname{cosec}\theta\) |
| III | − | − | \(\tan\theta,\ \cot\theta\) |
| IV | + | − | \(\cos\theta,\ \sec\theta\) |
Example: \(\tan\theta=-\dfrac9{40}\), \(90^\circ<\theta<180^\circ\)
\(\begin{aligned} x&=-40,\quad y=9 \\ r&=\sqrt{40^2+9^2}=41 \\ \sin\theta&=\dfrac{9}{41} \\ \cos\theta&=-\dfrac{40}{41} \\ \sec\theta&=-\dfrac{41}{40} \end{aligned}\)
Use the quadrant to check the sign of each ratio.
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