# Find the value of ${\sin ^{ - 1}}\{ \sin ( - {600^ \circ })\} $

Answer

Verified

381.9k+ views

Hint: To solve this problem we need to have basic knowledge about the trigonometric values, trigonometric identities and inverse trigonometric identities because the question here belongs to the inverse trigonometry concept.

Complete step-by-step answer:

Before solving this problem let us consider the given term as P.

Then $P = {\sin ^{ - 1}}\{ \sin ( - {600^ \circ })\} $

On using the trigonometric identity $\sin ( - \theta ) = - \sin \theta $ we can rewrite P as

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ({600^ \circ })\} $

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ({360^ \circ } \times 2 - {120^ \circ })\} $ $[\because {360^ \circ } = 2\pi ]$

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin (2\pi - {120^ \circ })\} $

Now by using the trigonometric identity $\sin (2\pi - A) = \sin ( - A)$ we can rewrite the above term as

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ( - {120^ \circ })\} $

Again by using the trigonometric identity $\sin ( - \theta ) = - \sin \theta $ we can rewrite the term as $

\Rightarrow {\sin ^{ - 1}}\{ - ( - \sin {120^ \circ })\} \\

\Rightarrow {\sin ^{ - 1}}\{ \sin ({120^ \circ })\} \\

$

Now this can also be written as

$ \Rightarrow {\sin ^{ - 1}}\{ \sin ({180^ \circ } - {60^ \circ })\} $

$ \Rightarrow {\sin ^{ - 1}}\{ \sin ({60^ \circ })\} $

On using the inverse trigonometric identity ${\sin ^{ - 1}}\{ \sin x\} = x$ we can rewrite the term as

$

\Rightarrow {60^ \circ } \\

\therefore P = {60^ \circ } \\

$

Hence the value of ${\sin ^{ - 1}}\{ \sin ( - {600^ \circ })\} = {60^ \circ }$

Note: The above solution is a step-by-step process of finding the value of a given term where we have included the trigonometric identities and inverse trigonometric identities to solve the question. This can also be done in a simple way, as $\sin ( - {600^ \circ }) = \sin ( - {600^ \circ } + {720^ \circ }) = \sin ({120^ \circ }) = \sin ({60^ \circ })$

Here finally we have found the theta value so the answer is $\theta = {60^ \circ }$.

Complete step-by-step answer:

Before solving this problem let us consider the given term as P.

Then $P = {\sin ^{ - 1}}\{ \sin ( - {600^ \circ })\} $

On using the trigonometric identity $\sin ( - \theta ) = - \sin \theta $ we can rewrite P as

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ({600^ \circ })\} $

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ({360^ \circ } \times 2 - {120^ \circ })\} $ $[\because {360^ \circ } = 2\pi ]$

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin (2\pi - {120^ \circ })\} $

Now by using the trigonometric identity $\sin (2\pi - A) = \sin ( - A)$ we can rewrite the above term as

$ \Rightarrow {\sin ^{ - 1}}\{ - \sin ( - {120^ \circ })\} $

Again by using the trigonometric identity $\sin ( - \theta ) = - \sin \theta $ we can rewrite the term as $

\Rightarrow {\sin ^{ - 1}}\{ - ( - \sin {120^ \circ })\} \\

\Rightarrow {\sin ^{ - 1}}\{ \sin ({120^ \circ })\} \\

$

Now this can also be written as

$ \Rightarrow {\sin ^{ - 1}}\{ \sin ({180^ \circ } - {60^ \circ })\} $

$ \Rightarrow {\sin ^{ - 1}}\{ \sin ({60^ \circ })\} $

On using the inverse trigonometric identity ${\sin ^{ - 1}}\{ \sin x\} = x$ we can rewrite the term as

$

\Rightarrow {60^ \circ } \\

\therefore P = {60^ \circ } \\

$

Hence the value of ${\sin ^{ - 1}}\{ \sin ( - {600^ \circ })\} = {60^ \circ }$

Note: The above solution is a step-by-step process of finding the value of a given term where we have included the trigonometric identities and inverse trigonometric identities to solve the question. This can also be done in a simple way, as $\sin ( - {600^ \circ }) = \sin ( - {600^ \circ } + {720^ \circ }) = \sin ({120^ \circ }) = \sin ({60^ \circ })$

Here finally we have found the theta value so the answer is $\theta = {60^ \circ }$.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

Which country launched the first satellite in space class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

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

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

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