Question

If 3cosa-4sina=5 then what is the value of 4cosa+3sina

Answer 1

see above pic for answer

Answer 2

Given:

3$\cos A$ - 4$\sin A$ = 5                    ............... (1)

Let 4$\cos A$ + 3$\sin A$ = x              ............... (2)

We have to find, the value of 4$\cos A$ + 3$\sin A$ = ?

Solution:

Squaring and adding equations (1) and (2), we get

$(3\cos A - 4\sin A)^2 + (4\cos A + 3\sin A)^2$ = $5^2$ + $x^{2}$    

⇒ 25 + $x^{2}$ = $(3\cos A)^2 + (4\sin A)^2 + 2(3\cos A)(4\sin A)+ (4\cos A)^2 + (3\sin A)^2-2(3\cos A)(4\sin A)$

⇒ 25 + $x^{2}$ = $9\cos^2 A +16\sin^2 A + 16\cos^2+ 9\sin^2 A$  

⇒ 25 + $x^{2}$ = $9(\cos^2 A+\sin^2 A) +16(\cos^2 A+\sin^2 A)$

Using the trigonometric identity:

$\cos^2 A$ + $\sin^2 A$ = 1

25 + $x^{2}$ = 9(1) + 16(1)

⇒ 25 + $x^{2}$ = 9 + 16

⇒ $x^{2}$ = 25 - 25

⇒ $x^{2}$ = 0

⇒ x = 0

∴ 4$\cos A$ + 3$\sin A$ = 0

Thus, if 3$\cos A$ - 4$\sin A$ = 5, then the value of "4$\cos A$ + 3$\sin A$ = 0".