Question

What is the exact value of sin 45° ?enter your answer, as a simplified fraction.

Answer 1

Look at the picture.

Use the Pythagorean theorem:

$b^2=a^2+a^2\\\\b^2=2a^2\to b=\sqrt{2a^2}\\\\b=\sqrt{a^2}\cdot\sqrt2\\\\b=a\sqrt2$

$\sin\theta=\dfrac{opposite}{hypotenuse}\\\\\theta=45^o,\ opposite=a,\ hypotenuse=a\sqrt2$

substitute

$\sin45^o=\dfrac{a}{a\sqrt2}=\dfrac{1}{\sqrt2}\cdot\dfrac{\sqrt2}{\sqrt2}=\dfrac{\sqrt2}{2}$

Answer: $\sin45^o=\dfrac{\sqrt2}{2}$