Mathematics: For questions 1 – 2, find an algebraic expression equivalent to the following expressions: 1. cos(tan^-1x) 2. cot(arcsin x)

Let x= distance to firetan29 = 24/xx tan29 =24x = 24/tan29x= 43.30m (2dp)...

For questions 1 – 2, find an algebraic expression

Kemby12

05.01.2022

Let $y=\tan^{-1}x$ , so that $\tan y=x$ . Picture a right triangle with hypotenuse of length and a reference angle . Then

$\cos y=\dfrac 1z$

We have

$\tan^2y=\sec^2y-1\implies x^2=z^2-1\implies z=\sqrt{x^2+1}$

$\implies\cos(\tan^{-1}x)=\cos y=\dfrac1{\sqrt{x^2+1}}$

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Similar strategy: Let $y=\arcsin x$ , so that $\sin y=x$ . In a right triangle with hypotenuse and reference angle , we have

$\cos y=\sqrt{1-x^2}[t/ex][tex]\implies\cot(\arcsin x)=\cot y=\dfrac{\cos y}{\sin y}=\dfrac{\sqrt{1-x^2}}x$

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