# How do you evaluate sin45°?

Jul 31, 2015

$\sin \left({45}^{o}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \ldots$

#### Explanation:

An angle of ${45}^{o}$ is an angle in the right triangle with equal catheti because another acute angel must be ${45}^{o}$ as well..
So, if a cathetus equals to $a$, another is $a$ as well and hypotenuse, according to Pythagorean Theorem, equals to $\sqrt{{a}^{2} + {a}^{2}} = a \cdot \sqrt{2}$.

Therefore, $\sin \left({45}^{o}\right)$, as the ratio of a cathetus opposite to this angle to a hypotenuse, equals to $\frac{a}{a \cdot \sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

As for evaluation of $\sin \left({45}^{o}\right)$, we just have to calculate the value of $\frac{\sqrt{2}}{2}$, which is approximately $0.7071 \ldots$