Question

If $3\sin A + 5\cos A = 5$, then value of ${\left( {3\cos A - 5\sin A} \right)^2}$ is
A.4
B.9
C.5
D.2

Answer 1

Here we will firstly find the expression from the given equation by squaring both sides of the given equation. Then we will assume the value of the expression ${\left( {3\cos A - 5\sin A} \right)^2}$ to be some variable. Then we will expand the equation and add both the equations obtained. We will then solve it to get the value of $x$.

Complete step-by-step answer:
Given is $3\sin A + 5\cos A = 5$.
Firstly we will square both side of the given equation to form an expression. Therefore, we get
$\Rightarrow {\left( {3\sin A + 5\cos A} \right)^2} = {5^2}$
Now we will apply the exponent on the expression by using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$. Therefore, we get
$\Rightarrow 9{\sin ^2}A + 25{\cos ^2}A + 2\left( {3\sin A} \right)\left( {5\cos A} \right) = 25$
$\Rightarrow 9{\sin ^2}A + 25{\cos ^2}A + 30\sin A\cos A = 25$……………………………..$\left( 1 \right)$
Now let us assume ${\left( {3\cos A - 5\sin A} \right)^2} = x$
Now we will apply the exponent on the expression by using the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. Therefore, we get
$\Rightarrow 9{\cos ^2}A + 25{\sin ^2}A - 30\sin A\cos A = x$……………………………..$\left( 2 \right)$
Now we will add the equation $\left( 1 \right)$ and the equation $\left( 2 \right)$. Therefore, we get
$\Rightarrow \left( {9{{\sin }^2}A + 25{{\cos }^2}A + 30\sin A\cos A} \right) + \left( {9{{\cos }^2}A + 25{{\sin }^2}A - 30\sin A\cos A} \right) = x + 25$
Now we will simplify this equation to get the value of $x$. Therefore, we get
$\Rightarrow 9\left( {{{\sin }^2}A + {{\cos }^2}A} \right) + 25\left( {{{\sin }^2}A + {{\cos }^2}A} \right) = x + 25$
We know the trigonometric property that ${\sin ^2}A + {\cos ^2}A = 1$. Therefore, we get
$\Rightarrow 9\left( 1 \right) + 25\left( 1 \right) = x + 25$
Multiplying the terms, we get
$\Rightarrow 9 + 25 = x + 25$
Adding and subtracting the like terms, we get
$\Rightarrow x = 9$
We know that the value of the ${\left( {3\cos A - 5\sin A} \right)^2}$ is equal to $x$.
Hence the value of ${\left( {3\cos A - 5\sin A} \right)^2}$ is equal to 9.
So, option B is the correct option.

Note: While simplifying the equation we have to apply some trigonometric identities in it to solve the equation. Trigonometric identities are only used in equations, where trigonometric functions are present. Here we should also know the basic algebraic identities. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation.