List the angles of the triangle in order from smallest to largest. In triangle Upper A Upper C Upper B, line segment Upper A Upper C has length 4.3, line segment Upper B Upper C has length 5.9, and line segment Upper B Upper A has length 3.5. A C B 5.9 4.3 3.5 Choose the correct order of the angles from smallest to largest.

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]

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