Question

1. Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given sides

`\begin{gathered} a=7.3in \\ b=13.2in \\ c=15.8in \end{gathered}`

STEP 2: Write the formula to calculate the missing angles

To get the missing angles, we use the cosine laws and sine laws stated below:

`\begin{gathered} a^2=b^2+c^2-2bc\cos A \\ b^2=a^2+c^2-2ac\cos B \\ c^2=a^2+b^2-2ab\cos C \\ \\ Sine\text{ law:} \\ (\sin A)/(a)=(\sin B)/(b)=(sinC)/(c) \end{gathered}`

STEP 3: Use the cosine law to find the angle A

`\begin{gathered} a^2=b^2+c^2-2bc\cos A \\ Making\text{ sin A the subject of the formula, we have:} \\ -2bc\cos A=a^2-b^2-c^2 \\ \sin A=(a^2-b^2-c^2)/(-2bc) \\ \\ By\text{ substitution,} \\ \cos A=(7.3^2-13.2^2-15.8^2)/(-2(13.2*15.8^)) \\ \cos A=(-370.59)/(-417.12)=0.888449367 \\ A=\cos^(-1)0.888449367 \\ A=27.3209628 \\ A\approx27.3^(\circ) \end{gathered}`

Angle A = 27.3 degrees

STEP 4: Find Angle B

Using sine rule, we have:

`\begin{gathered} (\sin A)/(a)=(\sin B)/(b) \\ By\text{ substitution,} \\ (\sin27.3)/(7.3)=(\sin B)/(13.2) \\ By\text{ cross multiplication,} \\ sinB*7.3=sin27.3*13.2 \\ sinB=(\sin27.3*13.2)/(7.3) \\ \sin B=0.82933892 \\ B=\sin^(-1)0.82933892 \\ B=56.0308892 \\ B=56.1^(\circ) \end{gathered}`

STEP 5: Calculate the Angle C

Recall that the sum of angles in a triangle is 180 degrees, therefore,

`\begin{gathered} A+B+C=180^(\circ) \\ By\text{ substitution,} \\ 27.3^(\circ)+56.1^(\circ)+C=180^(\circ) \\ C=180^(\circ)-27.3^(\circ)-56.1^(\circ) \\ C=180^(\circ)-83.4=96.6^(\circ) \end{gathered}`

Hence, the angles are:

`A=27.3^(\circ),B=56.1^(\circ),C=96.6^(\circ)`

OPTION A