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Home » Find the principal value of each of the following: (i) sec-1 (2 sin (3π/4)) (ii) sec-1 (2 tan (3π/4))
Find the principal value of each of the following: (i) sec-1 (2 sin (3π/4)) (ii) sec-1 (2 tan (3π/4))
CBSE | Class 12 | Exercise 4.4 | Inverse Trigonometric Functions | Maths | RD Sharma
Find the principal value of each of the following: (i) sec-1 (2 sin (3π/4)) (ii) sec-1 (2 tan (3π/4))

(iii) According to ques,

    \[se{{c}^{-1}}~\left( 2\text{ }sin\text{ }\left( 3\pi /4 \right) \right)\]

But since,

    \[~sin\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }1/\surd 2\]

Therefore,

    \[~2\text{ }sin\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }2\text{ }\times \text{ }1/\surd 2\]

    \[2\text{ }sin\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }\surd 2\]

Therefore by substituting above values in sec-1 (2 sin (3π/4)), we have

    \[Se{{c}^{-1~}}\left( \surd 2 \right)\]

Let,

    \[Se{{c}^{-1~}}\left( \surd 2 \right)\text{ }=\text{ }y\]

    \[Sec\text{ }y\text{ }=\text{ }\surd 2\]

    \[Sec\text{ }\left( \pi /4 \right)\text{ }=\text{ }\surd 2\]

Therefore range of principal value of sec-1 is:

    \[~\left[ 0,\text{ }\pi \right]\text{ }\text{ }\left\{ \pi /2 \right\}\]

and

 

    \[sec\text{ }\left( \pi /4 \right)\text{ }=\text{ }\surd 2\]

Therefore, the principal value of sec-1 (2 sin (3π/4)) is:

    \[~\pi /4.\]

(iv) According to ques,

    \[se{{c}^{-1}}~\left( 2\text{ }tan\text{ }\left( 3\pi /4 \right) \right)\]

But since,

    \[tan\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }-1\]

Therefore,

    \[2\text{ }tan\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }2\text{ }\times \text{ }-1\]

    \[2\text{ }tan\text{ }\left( 3\pi /4 \right)\text{ }=\text{ }-2\]

By substituting these values in sec-1 (2 tan (3π/4)), we get

    \[Se{{c}^{-1}}~\left( -2 \right)\]

Now let,

    \[~y\text{ }=\text{ }Se{{c}^{-1}}~\left( -2 \right)\]

    \[Sec\text{ }y\text{ }=\text{ }\text{ }2\]

    \[\text{ }sec\text{ }\left( \pi /3 \right)\text{ }=\text{ }-2\]

    \[=\text{ }sec\text{ }\left( \pi \text{ }\text{ }\pi /3 \right)\]

    \[=\text{ }sec\text{ }\left( 2\pi /3 \right)\]

Therefore,

the range of principal value of sec-1 is:

    \[~\left[ 0,\text{ }\pi \right]\text{ }\text{ }\left\{ \pi /2 \right\}\]

and

    \[sec\text{ }\left( 2\pi /3 \right)\text{ }=\text{ }-2\]

Therefore, the principal value of sec-1 (2 tan (3π/4)) is:

    \[\left( 2\pi /3 \right).\]

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