Answer

Verified

414k+ views

**Hint:**Convert the given angles of the inverse trigonometric functions so that their value lies in their domain and range and then substitute these values into the given expression. After the simplification, we have the result required in the given problem.

**Complete step-by-step answer:**

Consider the given expression in the problem:

${\sin ^{ - 1}}(\sin 12) + {\cos ^{ - 1}}(\cos 12)$

We have to find the value of the given trigonometric expression.

We know that the principal value of the inverse of sine angle is given as:

\[{\sin ^{ - 1}} \in \left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right]\], where the value of $x$ are \[x \in \left[ { - 1,1} \right]\].

Similarly, we know that the principal value of inverse of the cosine angle is given as:

${\cos ^{ - 1}}y \in \left[ {0,\pi } \right]$, where the value of $x$ are $y \in \left[ { - 1,1} \right]$

Now, in the term given to us is

${\sin ^{ - 1}}(\sin 12) \ne 12$, where $12 \notin \left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right]$

Similarly,${\cos ^{ - 1}}(\cos 12) \ne 12$, where$12 \notin \left[ {0,\pi } \right]$

Therefore, we can rewrite the given expression as:

${\sin ^{ - 1}}\left( {\sin 12} \right) = {\sin ^{ - 1}}\left( {\sin \left( {12 - 4\pi } \right)} \right)$

Let us rewrite the given function by changing the angles.

${\sin ^{ - 1}}(\sin 12) + {\cos ^{ - 1}}(\cos 12) = {\sin ^{ - 1}}\left( {\sin \left( {12 - 4\pi } \right)} \right) + {\cos ^{ - 1}}\left( {\cos \left( {4\pi - 12} \right)} \right)$

Then this obtained expression can be given as:

\[{\sin ^{ - 1}}(\sin 12) + {\cos ^{ - 1}}(\cos 12) = (12 - 4\pi ) + (4\pi - 12)\]

Usual formulas in the above expression as:

$\sin (2n\pi - x) = - \sin (x);$

$\sin ( - x) = - \sin x$

\[{\sin ^{ - 1}}(\sin 12) + {\cos ^{ - 1}}(\cos 12) = 0\]

So, we have the conclusion that:

\[{\sin ^{ - 1}}(\sin 12) + {\cos ^{ - 1}}(\cos 12) = 0\]

Therefore, the option (a) is correct.

**Note:**The domain of the function consists of all possible values of the independent variable where the function is defined and the range is the set which is obtained by the substitution of all values of the domain into the function.

The principal value of the inverse trigonometric function is the least absolute value of the angle.

Recently Updated Pages

If O is the origin and OP and OQ are the tangents from class 10 maths CBSE

Let PQ be the focal chord of the parabola y24ax The class 10 maths CBSE

Which of the following picture is not a 3D figure a class 10 maths CBSE

What are the three theories on how Earth was forme class 10 physics CBSE

How many faces edges and vertices are in an octagonal class 10 maths CBSE

How do you evaluate cot left dfrac4pi 3 right class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths