If-3sin-x-5cos-x-5-find-value-s-of-5sin-x-3cos-x-

Question Number 139710 by ajfour last updated on 30/Apr/21

I f 3 \sin x + 5 \cos x = 5

f i n d v a l u e (s) o f 5 \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 \lor \tan x = \frac{15}{8}

5 \sin x - 3 \cos x = - \frac{3 - 5 \tan x}{\sqrt{1 + \tan^{2} x}} = {\begin{cases} - 3 \\ 3 \end{cases}

Answered by john_santu last updated on 30/Apr/21

l e t 5 \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

(*) \sqrt{25 + 9} \cos (x - α) = 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 \dots (i)

5 \sin x - 3 \cos x = k \dots (i i)

{(i)}^{2} + {(i i)}^{2} :

3^{2} + 5^{2} = 5^{2} + k^{2}

\Rightarrow k = \pm 3

o r

\sqrt{3^{2} + 5^{2}} \sin (x + α) = 5

\Rightarrow \sin (x + α) = \frac{5}{\sqrt{3^{2} + 5^{2}}}

\Rightarrow \cos (x + α) = \pm \frac{3}{\sqrt{3^{2} + 5^{2}}}

k = \sqrt{3^{2} + 5^{2}} \cos (x + α) = \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 + 5 t}{34}

\cos x = \frac{25 - 3 t}{34}

25 {(3 + t)}^{2} + {(25 - 3 t)}^{2} = {(34)}^{2}

34 t^{2} + 25 \times 34 = 34 \times 34

\Rightarrow t^{2} = 9 \Rightarrow t = \pm 3

Answered by ajfour last updated on 30/Apr/21

T h a n k s b o t h S i r s .

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