How to Find the Length of Side in a Triangle Using the Law of Sines

What is the law of sines and how can it help us find the length of a side in a triangle?

Given the data in △abc where m∠a=39 °, a=11 , and b=13, how can we find side c to the nearest tenth?

Answer:

The length of side c in △ABC is found by utilizing the law of sines and simple geometry.

To find the length of side c in a triangle when given the measures of angles and sides, we can use a math theorem known as the law of sines. This law states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. Mathematically, the law of sines can be written as (a/sinA)=(b/sinB)=(c/sinC).

In our case, we are given that ∠a=39°, side a is 11 units long, and side b is 13 units long. To find the measure of side c, we first need to calculate the measure of ∠b. This can be done by subtracting the measure of ∠a from 180° since the sum of angles in a triangle is 180°. Therefore, ∠b = 180° - 39° = 141°.

Now that we have the measure of ∠b, we can use the law of sines to find the length of side c. By applying the formula c = (b*sinA)/sinB, we get c = (13* sin(39°))/sin(141°), which simplifies to approximately 9.2 units when rounded to the nearest tenth.

In conclusion, the law of sines is a powerful tool in trigonometry that allows us to find missing side lengths in a triangle when given specific angles and side lengths. By understanding and applying this theorem, we can solve geometric problems with ease and precision.