Express sin ^{ -1 } ((sinx + cosx)/√2); where (-π)/4 < 𝑥 < π/4 , in the simplest form.
CBSE Class 12 Sample Paper for 2020 Boards
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CBSE Class 12 Sample Paper for 2020 Boards
Last updated at April 16, 2024 by Teachoo
Question 21 (OR 1st Question) Express sin^(−1)((sin𝑥 + cos𝑥)/√2); where (−𝜋)/4 < 𝑥 < 𝜋/4 , in the simplest form. sin^(−1)((sin𝑥 + cos𝑥)/√2) = sin^(−1)(sin𝑥/√2+cos𝑥/√2) = sin^(−1)(sin𝑥×1/√2+cos𝑥×1/√2) = sin^(−1)(sin𝑥×cos〖𝜋/4〗+cos𝑥×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