Question

# If $\sin {\text{3}}A$ =$\cos \left( {A - {{26}^ \circ }} \right)$ where $3A$ is an acute angle, find the value of $A$.

Hint: In order to solve this question we have to convert $\sin$ in terms of $\cos {\text{ }}\left[ {{\text{As }}\sin \theta = \left( {\cos {{90}^ \circ } - \theta } \right)} \right]$. By doing so we will get both sides in terms of cosine.

Thus, it will be easier to solve this question.

Complementary Angles- Two angles are said to be complementary if their sum is equal to the right angle.

In this question we have given,

$\sin {\text{3}}A$ =$\cos \left( {A - {{26}^ \circ }} \right)$

In this question, we have to find out the value of $A$ .

As we know that

$\sin x = \cos ({90^ \circ } - x)$

So, by using the identity we can say ,

$\sin 3A = \cos \left( {{{90}^ \circ } - 3A} \right) = \cos \left( {A - {{26}^ \circ }} \right)$

Since, $3A$ is an acute angle we can say that ,

$\cos \left( {{{90}^ \circ } - 3A} \right) = \cos \left( {A - {{26}^ \circ }} \right)$

As we have $\cos$ on both sides , by eliminating cos we will get,

${90^ \circ } - 3A = A - {26^ \circ } \\ \Rightarrow 4A = {116^ \circ } \\ \Rightarrow {\text{ }}A = \dfrac{{{{116}^ \circ }}}{4} \\ \Rightarrow {\text{ }}A = {29^ \circ } \\$

Note: Whenever we face such types of questions, the key concept is that we must covert sin in terms of cos or vice versa. It is clearly visible that here $3A$ represents an acute angle. First we will use the identity $\sin \theta = \cos \left( {{{90}^ \circ } - \theta } \right)$ then eliminate cos (or sin) then by simplifying the equations we will get our required answer.