If $sin α = \frac{3}{5}$ $sinalpha=3/5$ and $α$ $alpha$ lies in Q2 and $sin β = \frac{5}{13}$ $sinbeta=5/13$ and $β$ $beta$ lies in Q1, find $cos (α + β)$ $cos(alpha+beta)$ , $cos β$ $cosbeta$ and $tan α$ $tanalpha$ ?

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If $sin α = \frac{3}{5}$ $sinalpha=3/5$ and $α$ $alpha$ lies in Q2 and $sin β = \frac{5}{13}$ $sinbeta=5/13$ and $β$ $beta$ lies in Q1, find $cos (α + β)$ $cos(alpha+beta)$ , $cos β$ $cosbeta$ and $tan α$ $tanalpha$ ?

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Answer 1 · 2017-03-16T16:16:09

$cos (α + β) = - \frac{63}{65}$ $cos(alpha+beta)=-63/65$ ; $cos β = \frac{12}{13}$ $cosbeta=12/13$ and $tan α = - \frac{3}{4}$ $tanalpha=-3/4$

Explanation:

As $sin α = \frac{3}{5}$ $sinalpha=3/5$ and $α$ $alpha$ lies in Q2, $cos α$ $cosalpha$ is negative and

$cos α = \sqrt{1 - {sin}^{2} α} = \sqrt{1 - {(\frac{3}{5})}^{2}}$ $cosalpha=sqrt(1-sin^2alpha)=sqrt(1-(3/5)^2)$

= $\sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = - \frac{4}{5}$ $sqrt(1-9/25)=sqrt(16/25)=-4/5$

and $tan α = \frac{sin α}{cos α} = \frac{\frac{3}{5}}{- \frac{4}{5}} = \frac{3}{5} \times (- \frac{5}{4}) = - \frac{3}{4}$ $tanalpha=sinalpha/cosalpha=(3/5)/(-4/5)=3/5xx(-5/4)=-3/4$

further as $sin β = \frac{5}{13}$ $sinbeta=5/13$ and $β$ $beta$ lies in Q1, $cos α$ $cosalpha$ is positive and

$cos β = \sqrt{1 - {sin}^{2} β} = \sqrt{1 - {(\frac{5}{13})}^{2}}$ $cosbeta=sqrt(1-sin^2beta)=sqrt(1-(5/13)^2)$

= $= \sqrt{1 - \frac{25}{169}} \sqrt{\frac{144}{25}} = \frac{12}{13}$ $=sqrt(1-25/169)sqrt(144/25)=12/13$

and $cos (α + β)$ $cos(alpha+beta)$

= $cos α cos β - sin α sin β$ $cosalphacosbeta-sinalphasinbeta$

= $- \frac{4}{5} \times \frac{12}{13} - \frac{3}{5} \times \frac{5}{13}$ $-4/5xx12/13-3/5xx5/13$

= $- \frac{48}{65} - \frac{15}{65} = - \frac{63}{65}$ $-48/65-15/65=-63/65$

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If $sin α = \frac{3}{5}$ $sinalpha=3/5$ and $α$ $alpha$ lies in Q2 and $sin β = \frac{5}{13}$ $sinbeta=5/13$ and $β$ $beta$ lies in Q1, find $cos (α + β)$ $cos(alpha+beta)$ , $cos β$ $cosbeta$ and $tan α$ $tanalpha$ ?

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Explanation:

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If $sin α = \frac{3}{5}$ $sinalpha=3/5$ and $α$ $alpha$ lies in Q2 and $sin β = \frac{5}{13}$ $sinbeta=5/13$ and $β$ $beta$ lies in Q1, find $cos (α + β)$ $cos(alpha+beta)$ , $cos β$ $cosbeta$ and $tan α$ $tanalpha$ ?