Jika tan α = 1, tan β = 1/3 dengan α dan β sudut lancip maka sin (α − β) =.... A. 2/3 √5 B. 1/5 √5 C.... Question from @aremania123

October 2018 2 11 Report

Jika tan α = 1, tan β = 1/3 dengan α dan β sudut lancip maka sin (α − β) =....
A. 2/3 √5
B. 1/5 √5
C. 1/2
D. 2/5
E. 1/5

wibowo1231 Dengan rumus selisih dua sudut:
$
sin ( \alpha - \beta ) sin \alpha cos \beta$
$= \frac{1}{ \sqrt{2} } . \frac{3}{ \sqrt{10} } - \frac{1}{ \sqrt{2} } . \frac{1}{ \sqrt{10} }$
$= \frac{2}{ \sqrt{20} } = \frac{2}{20} \sqrt{20} = \frac{2}{20} . 2 \sqrt{5} = \frac{1}{5} \sqrt{5}$

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izharulhaq78 $\tan\alpha=1\\\frac{a_1}{b_1}=1\\\\a_1=1\\b_1=1\\c_1=\sqrt{a_1^2+b_1^2}=\sqrt{1^2+1^2}=\sqrt{1+1}=\sqrt{2}\\\\\tan\beta=\frac{1}{3}\\\frac{a}{b}=\frac{1}{3}\\\\a_2=1\\b_2=3\\c_2=\sqrt{a_2^2+b_2^2}=\sqrt{1^2+3^2}=\sqrt{1+9}=\sqrt{10}\\\\\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\\\sin(\alpha-\beta)=\frac{a_1}{c_1}\times\frac{b_2}{c_2}-\frac{b_1}{c_1}\times\frac{a_2}{c_2}\\\sin(\alpha-\beta)=\frac{a_1b_2}{c_1c_2}-\frac{a_2b_1}{c_1c_2}$
$\sin(\alpha-\beta)=\frac{a_1b_2-a_2b_1}{c_1c_2}\\\sin(\alpha-\beta)=\frac{1\times3-1\times1}{\sqrt2\sqrt{10}}\\\sin(\alpha-\beta)=\frac{3-1}{\sqrt{20}}\\\sin(\alpha-\beta)=\frac{2}{2\sqrt{5}}\\\sin(\alpha-\beta)=\frac{1}{\sqrt{5}}\\\boxed{\sin(\alpha-\beta)=\frac{1}{5}\sqrt{5}}$

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