Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 14366 by RasheedSindhi last updated on 31/May/17

Determine a) cos(sin^(-1) x) b) sin(cos^(-1) x) c) cos^(-1) (sin x) d) sin^(-1) (cos x) e) sin^(-1) (sin x) f) sin(sin^(-1) x) g) sin^(-1) (sin^(-1) x) h) sin(sin x)

$$\mathrm{Determine} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{cos}\left(\mathrm{sin}^{-\mathrm{1}} \mathrm{x}\right) \\ $$$$\left.\mathrm{b}\right)\:\mathrm{sin}\left(\mathrm{cos}^{-\mathrm{1}} \mathrm{x}\right) \\ $$$$\left.\mathrm{c}\right)\:\mathrm{cos}^{-\mathrm{1}} \left(\mathrm{sin}\:\mathrm{x}\right) \\ $$$$\left.\mathrm{d}\right)\:\mathrm{sin}^{-\mathrm{1}} \left(\mathrm{cos}\:\mathrm{x}\right) \\ $$$$\left.\mathrm{e}\right)\:\mathrm{sin}^{-\mathrm{1}} \left(\mathrm{sin}\:\mathrm{x}\right) \\ $$$$\left.\mathrm{f}\right)\:\mathrm{sin}\left(\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}\right) \\ $$$$\left.\mathrm{g}\right)\:\mathrm{sin}^{-\mathrm{1}} \left(\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}\right) \\ $$$$\left.\mathrm{h}\right)\:\mathrm{sin}\left(\mathrm{sin}\:\mathrm{x}\right) \\ $$

Commented by Tinkutara last updated on 31/May/17

sin (sin^(−1) x) = x (Principal value in the interval [((−π)/2) , (π/2)])

$$\mathrm{sin}\:\left(\mathrm{sin}^{−\mathrm{1}} \:{x}\right)\:=\:{x}\:\left(\mathrm{Principal}\:\mathrm{value}\:\mathrm{in}\right. \\ $$$$\left.\mathrm{the}\:\mathrm{interval}\:\left[\frac{−\pi}{\mathrm{2}}\:,\:\frac{\pi}{\mathrm{2}}\right]\right) \\ $$

Commented by RasheedSindhi last updated on 01/Jun/17

Thank you.

$$\mathrm{Thank}\:\mathrm{you}. \\ $$

Answered by mrW1 last updated on 31/May/17

a) t=sin^(−1) x sin t=x cos (sin^(−1) x)=cos t=±(√(1−x^2 )) b) sin (cos^(−1) x)=±(√(1−cos (cos^(−1) x)^2 )) =±(√(1−x^2 )) c) t=sin x sin^(−1) t=x cos^(−1) (sin x)=cos^(−1) t=(π/2)−sin^(−1) t=(π/2)−x d) sin^(−1) (cos x)=(π/2)−cos^(−1) (cos x)=(π/2)−x e) sin^(−1) (sin x)=x f) sin (sin^(−1) x)=x g) sin^(−1) (sin^(−1) x) is defined when −1≤sin^(−1) x≤1 ⇒−sin(1)≤x≤sin (1)≈0.84 sin^(−1) (sin^(−1) x) ≈sin^(−1) ((x/(sin (1))))≈sin^(−1) ((x/(0.84))) h) sin(sin x) is defined for x∈R sin (sin x)≈sin (1)sin x≈0.84sin x

$$\left.{a}\right) \\ $$$${t}=\mathrm{sin}^{−\mathrm{1}} \:{x} \\ $$$$\mathrm{sin}\:{t}={x} \\ $$$$\mathrm{cos}\:\left(\mathrm{sin}^{−\mathrm{1}} {x}\right)=\mathrm{cos}\:{t}=\pm\sqrt{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$$$\left.{b}\right) \\ $$$$\mathrm{sin}\:\left(\mathrm{cos}^{−\mathrm{1}} {x}\right)=\pm\sqrt{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{cos}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} } \\ $$$$=\pm\sqrt{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$$$\left.{c}\right) \\ $$$${t}=\mathrm{sin}\:{x} \\ $$$$\mathrm{sin}^{−\mathrm{1}} {t}={x} \\ $$$$\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{sin}\:{x}\right)=\mathrm{cos}^{−\mathrm{1}} {t}=\frac{\pi}{\mathrm{2}}−\mathrm{sin}^{−\mathrm{1}} {t}=\frac{\pi}{\mathrm{2}}−{x} \\ $$$$\left.{d}\right) \\ $$$$\mathrm{sin}^{−\mathrm{1}} \:\left(\mathrm{cos}\:{x}\right)=\frac{\pi}{\mathrm{2}}−\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{cos}\:{x}\right)=\frac{\pi}{\mathrm{2}}−{x} \\ $$$$\left.{e}\right) \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:{x}\right)={x} \\ $$$$\left.{f}\right) \\ $$$$\mathrm{sin}\:\left(\mathrm{sin}^{−\mathrm{1}} {x}\right)={x} \\ $$$$\left.{g}\right) \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}^{−\mathrm{1}} {x}\right)\:{is}\:{defined}\:{when} \\ $$$$−\mathrm{1}\leqslant\mathrm{sin}^{−\mathrm{1}} {x}\leqslant\mathrm{1} \\ $$$$\Rightarrow−\mathrm{sin}\left(\mathrm{1}\right)\leqslant{x}\leqslant\mathrm{sin}\:\left(\mathrm{1}\right)\approx\mathrm{0}.\mathrm{84} \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}^{−\mathrm{1}} {x}\right)\:\approx\mathrm{sin}^{−\mathrm{1}} \left(\frac{{x}}{\mathrm{sin}\:\left(\mathrm{1}\right)}\right)\approx\mathrm{sin}^{−\mathrm{1}} \left(\frac{{x}}{\mathrm{0}.\mathrm{84}}\right) \\ $$$$\left.{h}\right) \\ $$$${sin}\left(\mathrm{sin}\:{x}\right)\:{is}\:{defined}\:{for}\:{x}\in{R} \\ $$$$\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\approx\mathrm{sin}\:\left(\mathrm{1}\right)\mathrm{sin}\:{x}\approx\mathrm{0}.\mathrm{84sin}\:{x} \\ $$

Commented by RasheedSindhi last updated on 01/Jun/17

Thanks Sir!

$$\mathcal{T}\mathrm{h}{a}\mathrm{n}{k}\mathrm{s}\:\mathrm{Sir}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com