Locate the quadrant and form a right-angled triangle. Use signed coordinates x,y and a positive radius r. r=sqrt(x2+y2) sintheta= yr costheta= xr tantheta= yx r>0. Tangent requires x 0; reciprocal ratios require a nonzero denominator. **Quadrant** | **x** | **y** | **Positive ratios** I | + | + | All II | − | + | sintheta, cosectheta III | − | − | tantheta, cottheta IV | + | − | costheta, sectheta Example: tantheta=- 940, 90 <theta<180 x=-40, y=9 r=sqrt(402+92)=41 sintheta= 941 costheta=- 4041 sectheta=- 4140 Use the quadrant to check the sign of each ratio.

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.

\(\sin\theta,\ \operatorname{cosec}\theta\)

Example: \(\tan\theta=-\dfrac9{40}\), \(90^\circ<\theta<180^\circ\)

Use the quadrant to check the sign of each ratio.

Ratios from a given quadrant

Ratios from a given quadrant

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