Express sin -1 ⁡((sin⁡x  + cos⁡x)/√2); where (-π)/4 < 𝑥 < π/4 , in the simplest form.

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  1. Class 12
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Transcript

Question 21 (OR 1st Question) Express sin^(−1)⁡((sin⁡𝑥 + cos⁡𝑥)/√2); where (−𝜋)/4 < 𝑥 < 𝜋/4 , in the simplest form. sin^(−1)⁡((sin⁡𝑥 + cos⁡𝑥)/√2) = sin^(−1)⁡(sin⁡𝑥/√2+cos⁡𝑥/√2) = sin^(−1)⁡(sin⁡𝑥×1/√2+cos⁡𝑥×1/√2) = sin^(−1)⁡(sin⁡𝑥×cos⁡〖𝜋/4〗+cos⁡𝑥×s𝑖𝑛⁡〖𝜋/4〗 ) We know that sin (A + B) = sin A cos B + cos A sin B = sin^(−1)⁡(sin⁡(𝑥+𝜋/4) ) = sin^(−1)⁡(sin⁡(𝑥+𝜋/4) ) Now, checking if angle is in principal value i.e. 0<𝑥+𝜋/4<𝜋/2 i.e. (−𝜋)/4 < 𝑥 < 𝜋/4 which is true So, we can write = 𝒙+𝝅/𝟒

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.