Question

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

Answer 1

See below

Explanation:

A right-angled triangle has three angles, `alpha`, `beta` and `theta`, with one of those being `90^"o"`. Let's say `theta = 90`.

Now, we're trying to prove that `sinalpha=cosbeta`. 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 `sinalpha=cos(90-alpha)`. Using the compound angle formula, `cos(a-b)=coscosb+sinasinb`, we can say that `cos(90-alpha)=cosalphacos90+sinalphasin90`.

`cos90=0, sin90=1therefore cosalphacos90-sinalphasin90=0cosalpha+1sinalpha=sinalpha`

`cos(90-alpha)=sinalpha`

`cos(90-alpha)=cosbeta`

`thereforecosbeta=sinalpha` ΟΕΔ