Question

Ai chứng minh giúp với: \(\left|m\cos\left(x\right)+\sin\left(x\right)\right|\le\sqrt{m^2+1}\)

Answer 1

Đặt \(\left|A\right|=\left|mcos\left(x\right)+sin\left(x\right)\right|\)

\(\Rightarrow A^2=\left[m.cos\left(x\right)+1.sin\left(x\right)\right]^2\le\left(m^2+1^2\right)\left[cos^2\left(x\right)+sin^2\left(x\right)\right]\)

\(\Rightarrow A^2\le m^2+1\) (Vì cos²x + sin²x = 1)

\(\Rightarrow\left|A\right|\le\sqrt{m^2+1}\) hay \(\left|mcos\left(x\right)+sin\left(x\right)\right|\le\sqrt{m^2+1}\) (đpcm)