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If \[\theta \] and\[\phi \] are acute angles such that \[\sin \theta = \dfrac{1}{2}\] and \[\cos \phi = \dfrac{1}{3}\], then \[\theta + \phi \] lies in
A.\[\left[ {\pi /3,\pi /2} \right]\]
B.\[\left[ {\pi /2,2\pi /3} \right]\]
C.\[\left[ {2\pi /3,5\pi /3} \right]\]
D.\[\left[ {5\pi /6,\pi } \right]\]

Last updated date: 13th Jun 2024
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Hint: In the given question, we have been given that the given angles are acute, i.e., less than \[90^\circ \]. Then, these angles are used as the arguments of the sine and cosine functions whose value is given. We have to calculate the sum of the two angles. We are going to check if the given value corresponds to any angle out of the ones \[\left( {0^\circ ,30^\circ ,45^\circ ,60^\circ {\rm{ }}or{\rm{ }}90^\circ } \right)\] given in the usual trigonometric value table. If it does, we are going to directly know the value of the angle. And if it does not, we will check for the range in which it lies.
Formula Used:
We are going to use the formula of the inverse of a trigonometric function, which is:
Let \[f\left( x \right)\] represent any trigonometric value, then if \[f\left( \theta \right) = A\], then
\[\theta = f'\left( A \right)\]

Complete step-by-step answer:
In the given question, the two given trigonometric values are \[\sin \theta = \dfrac{1}{2}\] and \[\cos \phi = \dfrac{1}{3}\].
Now, in the given question, both of the angles are acute, hence, less than \[90^\circ \]. Thus, the minimum inferable range of the values is \[\left[ {0,\dfrac{\pi }{2}} \right]\].
Now, we can solve for the value of \[\theta \] because \[\sin \dfrac{\pi }{6} = \dfrac{1}{2}\].
Hence, \[\theta = \dfrac{\pi }{6}\].
Now, the value of \[\cos \phi = \dfrac{1}{3}\] is not known conventionally. So, we are going to check in what range the value lies.
\[\dfrac{1}{3} = 0.33\]
Now, \[\cos \dfrac{\pi }{3} = \dfrac{1}{2} = 0.5\] and \[\cos \dfrac{\pi }{2} = 0\]
Hence, \[\phi \] lies between \[\dfrac{\pi }{3}{\rm{ and }}\dfrac{\pi }{2}\].
Now, \[\dfrac{\pi }{3} \le \phi \le \dfrac{\pi }{2}\]
Now, adding \[\theta \],
\[\dfrac{\pi }{3} + \theta \le \phi + \theta \le \dfrac{\pi }{2} + \theta \]
But, \[\theta = \dfrac{\pi }{6}\]
Hence, \[\dfrac{\pi }{3} + \dfrac{\pi }{6} \le \phi + \theta \le \dfrac{\pi }{2} + \dfrac{\pi }{6}\]
Thus, \[\dfrac{\pi }{2} \le \phi + \theta \le \dfrac{{2\pi }}{3}\]
Hence, the correct option is B.

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contain the known and the unknown and pick the one which is the most suitable and the most effective for finding the answer of the given question. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order, because even a slightest error is going to make the whole expression awry and is going to give us an incorrect answer.