Answer:
c ≈ 24.9, A ≈ 28.7°, B ≈ 57.3°
Step-by-step explanation:
in the picture
The measurement of c, ∠B, and ∠A are 24.9 units, 57.3°, and 28.7°, respectively.
The law of cosine helps us to know the third side of a triangle when two sides of the triangle are already known the angle opposite to the third side is given. It is given by the formula,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot \cos\theta}[/tex]
where
c is the third side of the triangle
a and b are the other two sides of the triangle,
and θ is the angle opposite to the third side, therefore, opposite to side c.
The sine rule of trigonometry helps us to equate the side of the triangles to the angles of the triangles. It is given by the formula,
[tex]\dfrac{\sin\ A}{\alpha} =\dfrac{\sin\ B}{\beta} =\dfrac{\sin\ C}{\gamma}[/tex]
where Sin A is the angle and α is the length of the side of the triangle opposite to angle A,
Sin B is the angle and β is the length of the side of the triangle opposite to angle B,
Sin C is the angle and γ is the length of the side of the triangle opposite to angle C.
Using the law of cosine, we write for c as,
[tex]c =\sqrt{a^2+b^2-2ab(\cos C)}\\\\c = \sqrt{12^2+21^2-2(12)(21)(\cos 94^o)}\\\\c = 24.9[/tex]
Now, using the sine law we can write,
[tex]\dfrac{\sin\ A}{a} =\dfrac{\sin\ B}{b} =\dfrac{\sin\ C}{c}[/tex]
[tex]\dfrac{\sin\ A}{12} =\dfrac{\sin\ B}{21} =\dfrac{\sin94}{24.9}[/tex]
B= Sin⁻¹ [(Sin 94°) × (21/24.9)]
B = 57.3°
A = Sin⁻¹ [(Sin 94°) × (12/24.9)]
A = 28.7°
Learn more about the Law of Cosine:
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