For a terminal ray through P(x,y)!= O, let r=sqrt(x2+y2)>0. Then sintheta=y/r, costheta=x/r, tantheta=y/x, cosectheta=r/y, sectheta=r/x, cottheta=x/y, whenever the denominator is nonzero. A full turn leaves the same ray. Scaling both signed coordinates by a positive number changes the triangle size but leaves every ratio unchanged. The radius is a length and stays positive in every quadrant. First find the signs of x and y. Calculate r, substitute into the six definitions, simplify exactly, then check each denominator. For P(-3,4), r=5: sintheta=4/5, costheta=-3/5, tantheta=-4/3, cosectheta=5/4, sectheta=-5/3, cottheta=-3/4. At P(0,2), tangent and secant are undefined, but cottheta=0/2=0. Do not infer that cotangent is undefined merely because tangent is undefined. O cannot define a terminal ray.

For a terminal ray through \(P(x,y)\ne O\), let \(r=\sqrt{x^2+y^2}>0\). Then \(\sin\theta=\frac{y}{r}\), \(\cos\theta=\frac{x}{r}\), \(\tan\theta=\frac{y}{x}\), \(\operatorname{cosec}\theta=\frac{r}{y}\), \(\sec\theta=\frac{r}{x}\), \(\cot\theta=\frac{x}{y}\), whenever the denominator is nonzero.

A full turn leaves the same ray. Scaling both signed coordinates by a positive number changes the triangle size but leaves every ratio unchanged. The radius is a length and stays positive in every quadrant.

First find the signs of x and y. Calculate r, substitute into the six definitions, simplify exactly, then check each denominator.

For \(P(-3,4)\), \(r=5\): \(\sin\theta=\frac{4}{5}\), \(\cos\theta=-\frac{3}{5}\), \(\tan\theta=-\frac{4}{3}\), \(\operatorname{cosec}\theta=\frac{5}{4}\), \(\sec\theta=-\frac{5}{3}\), \(\cot\theta=-\frac{3}{4}\).

At \(P(0,2)\), tangent and secant are undefined, but \(\cot\theta=\frac{0}{2}=0\). Do not infer that cotangent is undefined merely because tangent is undefined. \(O\) cannot define a terminal ray.

Signed coordinates and six ratios

Signed coordinates and six ratios