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

CBSE Class 12 Sample Paper for 2020 Boards
CBSE Class 12 Sample Paper for 2020 Boards
Last updated at Dec. 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