Answer:
[tex]C = 36.0[/tex]
[tex]B = 46.2[/tex]
[tex]A = 97.8[/tex]
Step-by-step explanation:
Given
[tex]\triangle ABC[/tex]
[tex]AC = 4.3[/tex]
[tex]BC = 5.9[/tex]
[tex]BA = 3.5[/tex]
Required
List the angles from smallest to largest
The given parameters is illustrated with the attached image.
[tex]AC = 4.3[/tex] -- b
[tex]BC = 5.9[/tex] --- a
[tex]BA = 3.5[/tex] --- c
This question will be solved using cosine rule
To calculate A, we have:
[tex]a^2 = b^2 + c^2 -2bc\ cos(A)[/tex]
So, we have:
[tex]5.9^2 = 4.3^2 + 3.5^2 - 2 * 4.3 * 3.5 * \cos(A)[/tex]
[tex]34.81 = 18.49+ 12.25 - 30.10* \cos(A)[/tex]
Collect like terms
[tex]34.81 - 18.49- 12.25 = - 30.10* \cos(A)[/tex]
[tex]4.07 = - 30.10* \cos(A)[/tex]
Make cos(A) the subject
[tex]\cos(A) = -\frac{4.07}{30.10}[/tex]
[tex]\cos(A) = -0.1352[/tex]
Take arccos of both sides
[tex]A = cos^{-1}(-0.1352)[/tex]
[tex]A = 97.8[/tex]
Solving for B, we have:
[tex]b^2 = a^2 + c^2 -2ac\ cos(B)[/tex]
This gives:
[tex]4.3^2 = 5.9^2 + 3.5^2 -2*5.9*3.5\ cos(B)[/tex]
[tex]18.49 = 34.81+ 12.25 -41.30 *\cos(B)[/tex]
Collect like terms
[tex]18.49 - 34.81 - 12.25 = -41.30 *\cos(B)[/tex]
[tex]-28.57 = -41.30 *\cos(B)[/tex]
Solve for cos(B)
[tex]\cos(B) = \frac{-28.57}{-41.30}[/tex]
[tex]\cos(B) = 0.6918[/tex]
Take arccos of both sides
[tex]B = cos^{-1}(0.6918)[/tex]
[tex]B = 46.2[/tex]
To solve for C, we make use of:
[tex]A + B + C = 180[/tex] --- angles in a triangle
[tex]97.8 + 46.2 + C = 180[/tex]
Collect like terms
[tex]C = - 97.8 - 46.2 + 180[/tex]
[tex]C = 36.0[/tex]
So, we have:
[tex]C = 36.0[/tex]
[tex]B = 46.2[/tex]
[tex]A = 97.8[/tex]
