Find the missing side or angle. Round to the nearest tenth. A=15° C= 120° b=3 c=[? ] Enter

Find the missing side or angle.
Round to the nearest tenth.
A=15°
C= 120°
b=3
c=[? ]
Enter

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  1. Given:

    [tex]A=15^\circ, C=120^\circ,b=3[/tex].

    To find:

    The length of side c.

    Solution:

    According to the angle sum of property of a triangle, the sum of all interior angles of a triangle is 180 degrees.

    [tex]m\angle A+m\angle B+m\angle C=180^\circ[/tex]

    [tex]15^\circ+m\angle B+120^\circ=180^\circ[/tex]

    [tex]m\angle B+135^\circ=180^\circ[/tex]

    [tex]m\angle B=180^\circ-135^\circ[/tex]

    [tex]m\angle B=45^\circ[/tex]

    According to the Law of sines,

    [tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

    Using Law of sines, we get

    [tex]\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

    [tex]\dfrac{3}{\sin 45^\circ}=\dfrac{c}{\sin 120^\circ}[/tex]

    [tex]\dfrac{3}{\dfrac{1}{\sqrt{2}}}=\dfrac{c}{\dfrac{\sqrt{3}}{2}}[/tex]

    [tex]3\sqrt{2}\times \dfrac{\sqrt{3}}{2}=c[/tex]

    On further simplification, we get

    [tex]\dfrac{3\sqrt{6}}{2}=c[/tex]

    [tex]3.674235=c[/tex]

    [tex]c\approx 3.7[/tex]

    Therefore, the length of side c is about 3.7 units.

    Reply

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