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 139710 by ajfour last updated on 30/Apr/21

$$If3\sin x+5\cos x=5$$$$findvalue\left(s\right)of5\sin x-3\cos x.$$

Answered by MJS_new last updated on 30/Apr/21

$$3\sin x+5\cos x=5$$$$\frac{3\tan x+5}{\sqrt{1+{\tan }^{2}x}}=5\Rightarrow \tan x=0\vee \tan x=\frac{15}{8}$$$$5\sin x-3\cos x=-\frac{3-5\tan x}{\sqrt{1+{\tan }^{2}x}}=\{\begin{array}{l}-3\\ 3\end{array}$$

Answered by john_santu last updated on 30/Apr/21

$$let5\sin x-3\cos x=p$$$$\Rightarrow 9\sin {}^{2}x+30\sin x\cos x+25\cos {}^{2}x=25$$$$\Rightarrow 9\cos {}^{2}x-30\sin x\cos x+25\sin {}^{2}x=p^2$$$$\Rightarrow 9+25=p^2+25$$$$\Rightarrow p^2=9;p=\pm 3$$$$(\ast )\sqrt{25+9}\cos (x-\alpha )=p$$$$\Rightarrow -1⩽\frac{p}{\sqrt{34}}⩽1$$

Answered by mr W last updated on 30/Apr/21

$$3\sin x+5\cos x=5...\left(i\right)$$$$5\sin x-3\cos x=k...\left(ii\right)$$$${\left(i\right)}^2+{\left(ii\right)}^2:$$$$3^2+5^2=5^2+k^2$$$$\Rightarrow k=\pm 3$$$$$$$$or$$$$\sqrt{3^2+5^2}\sin (x+\alpha )=5$$$$\Rightarrow \sin (x+\alpha )=\frac{5}{\sqrt{3^2+5^2}}$$$$\Rightarrow \cos (x+\alpha )=\pm \frac{3}{\sqrt{3^2+5^2}}$$$$k=\sqrt{3^2+5^2}\cos (x+\alpha )=\pm 3$$

Answered by ajfour last updated on 30/Apr/21

$$3\sin x+5\cos x=5$$$$5\sin x-3\cos x=t$$$$\sin x=\frac{15+5t}{34}$$$$\cos x=\frac{25-3t}{34}$$$$25{(3+t)}^2+{(25-3t)}^2={\left(34\right)}^2$$$$34t^2+25\times 34=34\times 34$$$$\Rightarrow t^2=9\Rightarrow t=\pm 3$$

Answered by ajfour last updated on 30/Apr/21

$$ThanksbothSirs.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com