Question

find sin theta such that 3cos theta +4sin theta​= 4

Answer 1

Answer:

$\bold\red{\sin\theta=1\:and\:\frac{7}{25}}$

Step-by-step explanation:

For simplicity, let's denote theta as alpha.

Now, According to Question,

$3 \cos( \alpha ) + 4 \sin( \alpha ) = 4$

But, we know that,

$\cos( \alpha ) = \sqrt{1 - { (\sin\alpha })^{2} }$

so putting this value , we get

further simplifying,

we get,

$= > 9(1 - { \sin}^{2} \alpha ) = 16(1 + { \sin }^{2} \alpha - 2 \sin( \alpha ) ) \\ \\ = > (16 + 9) { \sin }^{2} \alpha - 32 \sin( \alpha ) + (16 - 9) = 0 \\ \\ = > 25 { \sin }^{2} \alpha - 32 \sin( \alpha ) + 7 = 0 \\ \\ = > 25 { \sin }^{2} \alpha - 25 \sin( \alpha) - 7 \sin( \alpha ) + 7 = 0 \\ \\ = > 25 \sin \alpha ( \sin\alpha - 1) - 7( \sin \alpha - 1) = 0 \\ \\ = > ( \sin \alpha - 1)(25 \sin\alpha - 7) = 0 \\ \\ = > \sin( \alpha ) = 1 \: \: \: \: and \: \: \: \: \frac{7}{25}$

Hencw, the value of sin theta is 1 and 7/25.