10.1 Non-right triangles: law of sines  (Page 2/10)

The three angles must add up to 180 degrees. From this, we can determine that

To find an unknown side, we need to know the corresponding angle and a known ratio. We know that angle $\alpha =\mathrm{50°}$ and its corresponding side $a=10.$ We can use the following proportion from the Law of Sines to find the length of $c.$

Similarly, to solve for $b,$ we set up another proportion.

Therefore, the complete set of angles and sides is

Use the Law of Sines to find angle $\beta$ and angle $\gamma ,$ and then side $c.$ Solving for $\beta ,$ we have the proportion

However, in the diagram, angle $\beta$ appears to be an obtuse angle and may be greater than 90°. How did we get an acute angle, and how do we find the measurement of $\beta ?$ Let’s investigate further. Dropping a perpendicular from $\gamma$ and viewing the triangle from a right angle perspective, we have [link] . It appears that there may be a second triangle that will fit the given criteria.

The angle supplementary to $\beta$ is approximately equal to 49.9°, which means that $\beta =\mathrm{180°}-\mathrm{49.9°}=\mathrm{130.1°}.$ (Remember that the sine function is positive in both the first and second quadrants.) Solving for $\gamma ,$ we have

We can then use these measurements to solve the other triangle. Since ${\gamma }^{\prime }$ is supplementary to the sum of ${\alpha }^{\prime }$ and ${\beta }^{\prime },$ we have

Now we need to find $c$ and $c^{\prime }.$

We have

Finally,

To summarize, there are two triangles with an angle of 35°, an adjacent side of 8, and an opposite side of 6, as shown in [link] .

However, we were looking for the values for the triangle with an obtuse angle $\beta .$ We can see them in the first triangle (a) in [link] .