Question

Cho biết \(\sin\alpha+\cos\alpha=\sqrt{2}\).Tính \(\sin\alpha\) và \(\cos\alpha\)

Answer 1

đặt \(\sin a=\dfrac{x}{y};\cos a=\dfrac{z}{y}\)

Ta có : \(\sin a+\cos a=\sqrt{2}\)

\(\dfrac{x}{y}+\dfrac{z}{y}=\sqrt{2}\Rightarrow\dfrac{x+z}{y}=\sqrt{2}\)

\(\Rightarrow\dfrac{\left(x+z\right)^2}{y^2}=2\)

Mà ta có: x²+z²=y²(pitago)

\(\Rightarrow\dfrac{x^2+z^2+2xz}{x^2+z^2}=2\Rightarrow1+\dfrac{2xz}{x^2+z^2}=2\)

\(\Rightarrow\dfrac{2xz}{x^2+z^2}=1\Rightarrow2xz=x^2+z^2\Rightarrow x=z\)

=>\(a=45^o\)

=>\(\sin a=\sin45=\dfrac{\sqrt{2}}{2}\)

\(\cos a=\cos45=\dfrac{\sqrt{2}}{2}\)