Question

Solve: ${{\sin }^{2}}{{60}^{\circ }}+{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)=1$ .

Answer 1

Hint: At first take the term ${{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)$ from left to right hand side so we get, ${{\sin }^{2}}{{60}^{\circ }}=1-{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)$ after that we will use the identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ and use it as $1-{{\cos }^{2}}\theta ={{\sin }^{2}}\theta$ and then take $\theta$ as $3x-{{9}^{\circ }}$ and then further solve it to find value of x.

Complete step-by-step answer:
In the question we are given the equation ${{\sin }^{2}}{{60}^{\circ }}+{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)=1$ and we have to find the value of x such that it satisfies the equation.
So, we are given the equation, ${{\sin }^{2}}{{60}^{\circ }}+{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)=1$ .
Now we will take ${{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)$ left to right hand side so we get,
${{\sin }^{2}}{{60}^{\circ }}=1-{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)$ .
We all know the identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ .
So, we can also write it as ${{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta$ now will take $\theta$ as $3x-{{9}^{\circ }}$ so we get,
${{\sin }^{2}}{{60}^{\circ }}=1-{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)$ or, ${{\sin }^{2}}{{60}^{\circ }}={{\sin }^{2}}\left( 3x-{{9}^{\circ }} \right)$ .
So from this we can write that ${{60}^{\circ }}=3x-{{9}^{\circ }}$ .
Now subtracting $3x$ from both the sides we get,
${{60}^{\circ }}-3x=3x-{{9}^{\circ }}-3x$ or, ${{60}^{\circ }}-3x=-{{9}^{\circ }}$ .
Now subtracting ${{60}^{\circ }}$ from both the sides we get,
${{60}^{\circ }}-3x-{{60}^{\circ }}=-{{9}^{\circ }}-{{60}^{\circ }}$ or, $-3x=-69$ .
Hence the value of x will be $\dfrac{-69}{-3}$ or, 23
So, the answer is 23.

Note: Also we can do this question by another way by directly using the identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ and compare it with ${{\sin }^{2}}{{60}^{\circ }}+{{\cos }^{2}}\left( 3x-{{9}^{\circ }} \right)=1$ . Then we can say that $3x-{{9}^{\circ }}={{60}^{\circ }}$ and then solve it to get the value of x.