IB AA SL: core and optional content Core: coordinate forms of sin, cos and tan, and inverse notation used by a calculator. Optional: reciprocal-function definitions and their coordinate forms. \(\sec\theta=\dfrac1{\cos\theta}\qquad\operatorname{cosec}\theta=\dfrac1{\sin\theta}\qquad\cot\theta=\dfrac{\cos\theta}{\sin\theta}\) Each denominator must be nonzero. Cotangent equals 1/tan θ only where both expressions exist. For \(P(x,y)\ne O\), let \(r=\sqrt{x^2+y^2}>0\). RatioCoordinate form \(\sin\theta\)\(y/r\) \(\cos\theta\)\(x/r\) \(\tan\theta\)\(y/x\), \(x\ne0\) \(\operatorname{cosec}\theta\)\(r/y\), \(y\ne0\) \(\sec\theta\)\(r/x\), \(x\ne0\) \(\cot\theta\)\(x/y\), \(y\ne0\) Powers and inverse functions \(\cos^n\theta=(\cos\theta)^n\), and similarly for sine and tangent, for positive integers \(n\). \(\cos^{-1}x\) denotes the inverse function, not \(1/\cos x\). The same convention applies to sine and tangent. \(y=\cos^{-1}x\iff\cos y=x\text{ and}\hspace{0.5em}0\leq y\leq\pi\), for \(-1\leq x\leq1\). \(y=\sin^{-1}x\iff\sin y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\), for \(-1\leq x\leq1\). \(y=\tan^{-1}x\iff\tan y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}<y<\frac{\pi}{2}\), for real \(x\).

Cosecant, Secant and Cotangent Ratios

IB AA SL: core and optional content

Core: coordinate forms of sin, cos and tan, and inverse notation used by a calculator. Optional: reciprocal-function definitions and their coordinate forms.

\(\sec\theta=\dfrac1{\cos\theta}\qquad\operatorname{cosec}\theta=\dfrac1{\sin\theta}\qquad\cot\theta=\dfrac{\cos\theta}{\sin\theta}\)

Each denominator must be nonzero. Cotangent equals 1/tan θ only where both expressions exist.

For \(P(x,y)\ne O\), let \(r=\sqrt{x^2+y^2}>0\).

Ratio	Coordinate form
\(\sin\theta\)	\(y/r\)
\(\cos\theta\)	\(x/r\)
\(\tan\theta\)	\(y/x\), \(x\ne0\)
\(\operatorname{cosec}\theta\)	\(r/y\), \(y\ne0\)
\(\sec\theta\)	\(r/x\), \(x\ne0\)
\(\cot\theta\)	\(x/y\), \(y\ne0\)

Powers and inverse functions

\(\cos^n\theta=(\cos\theta)^n\), and similarly for sine and tangent, for positive integers \(n\).

\(\cos^{-1}x\) denotes the inverse function, not \(1/\cos x\). The same convention applies to sine and tangent.

\(y=\cos^{-1}x\iff\cos y=x\text{ and}\hspace{0.5em}0\leq y\leq\pi\), for \(-1\leq x\leq1\).

\(y=\sin^{-1}x\iff\sin y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\), for \(-1\leq x\leq1\).

\(y=\tan^{-1}x\iff\tan y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}<y<\frac{\pi}{2}\), for real \(x\).

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