Question

Determine whether the given information results in one, two, or three triangles. Solve for any triangles. a=6, c=4, C=10

Answer 1

Given the question in the question tab, the following are the solution steps to get the answer

Step 1: Write the given sides and angles

By substitution, we have

`\begin{gathered} (6)/(\sin A)=(4)/(\sin10) \\ By\text{ cross multiplication, we have} \\ 6*\sin 10=4*\sin A \\ \sin A=(6*\sin10)/(4) \\ A=\sin ^(-1)((6*\sin10)/(4)) \\ A=15.09808661 \\ A\approx15^(\circ) \\ \text{This makes Angle B to be} \\ B=180-10+15(sum\text{ of angles in a triangle equals 180)} \\ B=180-25 \\ B=155^(\circ) \end{gathered}`

Step 3: To calculate length b using the sine rule;

`\begin{gathered} (b)/(\sin B)=(c)/(\sin C) \\ (b)/(\sin155)=(4)/(\sin 10) \\ b=(4*\sin 155)/(\sin 10) \\ b=9.735046285 \\ b\approx9.7 \end{gathered}`

Therefore the sides of the first triangle becomes 6, 9.7 and 10 units while it’s angles are 15,10 and 155 degrees.

Step 4: Determine if there is another triangle

Also, if two sides are given as 6 units and 4 units respectively, then the third side can be calculated by using the Pythagoras theorem and this immediately presumes that it’s a right angled triangle (one of the angles equals 90 degrees). The theorem states that;

`hyp^2=adj^2+opp^2`

Taking the other two as 6 and 4 units, the formula now becomes;

`\begin{gathered} b^2=a^2+c^2 \\ b^2=6^2+4^2 \\ b^2=36+16 \\ b=\sqrt[]{42} \\ b=6.480740698 \\ b\approx6.48^(\circ) \end{gathered}`

Therefore the sides of the second triangle becomes 6, 6.48 and 4 units with angles as 10, 80 and 90 degrees.