# A right-angled triangle has three angles, alpha, beta and theta with theta=90. How do I prove that sinalpha=cosbeta?

Mar 31, 2017

See below

#### Explanation:

A right-angled triangle has three angles, $\alpha$, $\beta$ and $\theta$, with one of those being ${90}^{\text{o}}$. Let's say $\theta = 90$.

Now, we're trying to prove that $\sin \alpha = \cos \beta$. Since all the angles in a triangle add up to $180$, and one of these angles is already $90$, then we can say that $\alpha = 90 - \beta$ and $\beta = 90 - \alpha .$

Thus $\sin \alpha = \cos \left(90 - \alpha\right)$. Using the compound angle formula, $\cos \left(a - b\right) = \cos \cos b + \sin a \sin b$, we can say that $\cos \left(90 - \alpha\right) = \cos \alpha \cos 90 + \sin \alpha \sin 90$.

$\cos 90 = 0 , \sin 90 = 1 \therefore \cos \alpha \cos 90 - \sin \alpha \sin 90 = 0 \cos \alpha + 1 \sin \alpha = \sin \alpha$

$\cos \left(90 - \alpha\right) = \sin \alpha$

$\cos \left(90 - \alpha\right) = \cos \beta$

$\therefore \cos \beta = \sin \alpha$ ΟΕΔ