In a \triangle ABC,\frac{a}{b}=2+\sqrt{3} and \angle C=60^\circ Then find the

Esther Hoffman

Esther Hoffman

Answered question

2022-04-24

In a ABC,ab=2+3 and C=60 Then find the ordered pairs (A,B)=

Answer & Explanation

Kathleen Keller

Kathleen Keller

Beginner2022-04-25Added 20 answers

sine rule
sinAa=sinBb=sinCc
ab=sinAsinB=2+3
A+B=180C=120
B=120A
sinAsinB=2+3
sinAsin(120A)=2+3
now solve for A
sinAsin120cosAcos120sinA=2+3
but it seem to solve for B is easier !
sin(120B)sinB=2+3
32cosB12sinBsinB=2+3
32cotB12=2+3
Aliana Porter

Aliana Porter

Beginner2022-04-26Added 6 answers

Start with Sine Law since you are given the quotient of sides.
Make a rough sketch of segment containing an angle 60, to note that A must be > 90 an obtuse angle anyhow, B is noted to be a narrow angle, should be an acute angle.
Arguments in degrees to avoid Latex.
sinAsin(A+60)=2+31, recognize the right hand side as tan75.
So, using sine supplementary angle identiity sinθ=sin(180θ)
sinAsin(A+60)=sin75cos75=sin75sin15=sin105sin165
=sin(105)sin(105+60)
Comparing arguments of first and last fractions we straightaway note that A=105
(A,B,C)=(105,15,60)

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