A right-angled triangle has three angles, $α$ $alpha$ , $β$ $beta$ and $θ$ $theta$ with $θ = 90$ $theta=90$ . How do I prove that $sin α = cos β$ $sinalpha=cosbeta$ ?

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A right-angled triangle has three angles, $α$ $alpha$ , $β$ $beta$ and $θ$ $theta$ with $θ = 90$ $theta=90$ . How do I prove that $sin α = cos β$ $sinalpha=cosbeta$ ?

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Answer 1 · 2017-03-31T14:52:19

See below

Explanation:

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

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

Thus $sin α = cos (90 - α)$ $sinalpha=cos(90-alpha)$ . Using the compound angle formula, $cos (a - b) = cos cos b + sin a sin b$ $cos(a-b)=coscosb+sinasinb$ , we can say that $cos (90 - α) = cos α cos 90 + sin α sin 90$ $cos(90-alpha)=cosalphacos90+sinalphasin90$ .

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

$cos(90-alpha)=sinalpha$

$cos(90-alpha)=cosbeta$

$thereforecosbeta=sinalpha$ ΟΕΔ

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A right-angled triangle has three angles, $α$ $alpha$ , $β$ $beta$ and $θ$ $theta$ with $θ = 90$ $theta=90$ . How do I prove that $sin α = cos β$ $sinalpha=cosbeta$ ?

1 Answer

Explanation:

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A right-angled triangle has three angles, alphaαalpha, betaβbeta and thetaθtheta with theta=90θ=90theta=90. How do I prove that sinalpha=cosbetasinα=cosβsinalpha=cosbeta?

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Explanation:

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A right-angled triangle has three angles, $α$ $alpha$ , $β$ $beta$ and $θ$ $theta$ with $θ = 90$ $theta=90$ . How do I prove that $sin α = cos β$ $sinalpha=cosbeta$ ?