Question

If $\cos \left( {20^\circ + \theta } \right) = \sin 30^\circ ,$ then the value of $\theta$ is :
(A) $20^\circ$
(B) $50^\circ$
(C) $30^\circ$
(D) $40^\circ$

Answer 1

In this first we will convert $\sin$ into $\cos$ on the right hand side by using complementary angle ratios. After converting we see that we can cancel out $\cos$ from both the sides. Now, after cancelling $\cos$ from both the sides only angles will remain on both the sides. Now, we just have to simplify the equation we got.

Complete step by step solution: Given, $\cos \left( {20^\circ + \theta } \right) = \sin 30^\circ$
Now, we will convert $\sin$ into $\cos$ on the right hand side of the above equation. Therefore, the equation can be written as:
$\cos \left( {20^\circ + \theta } \right) = \cos \left( {90^\circ - 30^\circ } \right)$
Now, the above equation can be written as:
$\Rightarrow \cos \left( {20^\circ + \theta } \right) = \cos \left( {60^\circ } \right)$
Now, we can cancel out $\cos$ on both the side of the above equation. Now, the above equation can be written as:
$\Rightarrow \left( {20^\circ + \theta } \right) = \left( {60^\circ } \right)$
Now, simplify the above equation:
$\Rightarrow \theta = 60^\circ - 20^\circ = 40^\circ$

Hence, the correct option is (D).

Additional information: We should know that we can write $\cos \left( {90^\circ - \theta } \right) = \sin \theta$ and we can also write $\sin \left( {90 - \theta } \right) = \cos \theta$. These conversions are important to solve these types of questions.

Note: The important thing in this question is the conversion on the right hand side of the equation because it will help in simplifying the equation. And then we just have to subtract the angle which we got. So, just be careful while converting because it will solve half of the question and will lead to wrong answers if done incorrect.