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Example

Solve completely the ∆ABC in which a = 12.4cm, c = 14.7cm and C = 72.1°

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Solution:

To ‘solve completely’ means to find all the unknown lengths and angles.

Screenshot 2022 06 08 at 01.19.06

Using the sine rule:

\( \frac{SinA}{a} = \frac{SinC}{c} \)

\( \frac{SinA}{12.4} = \frac{Sin72.1}{14.7} \)

SinA = \( \frac{12.4 \; \times \; Sin72.1}{14.7}\)

SinA = \( \frac{12.4 \; \times \; 0.9516}{14.7}\)

SinA = \( \frac{11.79984}{14.7} = \scriptsize 0.80271\)

A = Sin-1(0.80271)

A = 53.38 ≅ 53.4°

\( \frac{SinB}{b} = \frac{c}{SinC} \)

Sum of angles of ∆ = 180

∴ A + B +C = 180°

53.4 + B + 72.1 = 180

B = 180 – 125.5

B = 54.5°

Thus \( \frac{b}{Sin54. 5} = \frac{14.7}{Sin72.1} \)

b = \( \frac {14.7 \: \times \: Sin54.5}{Sin72.7}\)

b = \( \frac{11.9675}{0.9516} \scriptsize = 12.57 \)

b≅ 12.6cm

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