Question

if 3sin (theta) +4 cos (theta) =5 then find the value of 3 cos (theta) - 4 sin (theta)

Answer 1

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Answer 2

The value of $3 \cos \theta-4 \sin \theta=0$

Solution:

Given that:

$3 \sin \theta+4 \cos \theta=5$

To find : $3 \cos \theta-4 \sin \theta$

Let us use the given information and solve

$3 \sin \theta+4 \cos \theta=5$

On dividing the complete equation by 5 we get,

$\frac{3}{5} \sin \theta+\frac{4}{5} \cos \theta=1$  ------ eqn 1

Comparing with the known identity:

$\sin x \cos y+\cos x \sin y=\sin (x+y)$  ---- eqn 2

On comparing (1) and (2), we get

$\cos y=\frac{3}{5} \text { and } \sin y=\frac{4}{5} \text { and } x=\theta \text { and } \sin (x+y)=1$

$y=\sin ^{-1} \frac{4}{5}=53.13=53$

So, sin(x + y ) = 1

Substitute x = θ and y = 53

$\begin{array}{l}{\sin \left(53^{\circ}+\theta\right)=1} \\\\ {53^{\circ}+\theta=\sin ^{-1}(1)}\end{array}$

$\text { since } \sin 90^{\circ}=1, \text { hence } \sin ^{-1}(1)=90^{\circ}$

$\begin{array}{l}{\text {Therefore } 53^{\circ}+\theta=90^{\circ}} \\\\ {\theta=90-53=37^{\circ}}\end{array}$

Now, the value of:

$\begin{array}{l}{3 \cos \theta-4 \sin \theta=-3 \times \cos \left(37^{\circ}\right)-4 \times \sin \left(37^{\circ}\right)} \\\\ {=3 \times \frac{4}{5}-4 \times \frac{3}{5}=0}\end{array}$

Thus the value of $3 \cos \theta-4 \sin \theta=0$

Learn more about trignometric identities

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