Find all the angles and lengths of the given triangle, giving the lengths to three decimal places and the angles to the nearest degree.
Since we don’t have a right triangle, we can’t use the Pythagorean theorem and we can’t use our simple sine, cosine, and tangent. However, we can use the sine rule. The sine rule states that lowercase 𝑎 divided by the sin of uppercase 𝐴 is equal to lowercase 𝑏 divided by the sin of uppercase 𝐵 is equal to lowercase 𝑐 divided by the sine of uppercase 𝐶.
The lowercase letters 𝑎, 𝑏, and 𝑐 are sides, and the capital letters, the uppercase letters, are the angles, so here we already have in our triangle the capital letters, the uppercase letters, which are our angles. So the lowercase letters are located directly across from our up- uppercase letters.
So lowercase 𝑎 would be located on the side 𝐵𝐶 which is 6.3, lowercase 𝑏 would be the side 𝐴𝐶 which is across from uppercase 𝐵, and then lowercase 𝑐 would be the side 𝐴𝐵 which is across from the uppercase capital 𝐶.
So let’s go ahead and fill in what we know. We know capital 𝐴 is 41 degrees, we know capital 𝐵 is equal to 82 degrees, and we know lowercase 𝑎 is equal to 6.3. So we need to find lowercase 𝑏 and 𝑐 and then capital 𝐶. So we can set up proportions, cross-multiply, and solve for whatever that we’re missing.
Let’s first begin with finding 𝑏, lowercase 𝑏. So the first thing we should do is to find the cross-product; we cross-multiply. So we take 6.3 times the sin of 82 and set it equal to 𝑏 times the sin of 41. So to solve for 𝑏, let’s divide by the sin of 41, so they cancel on the right-hand side, so we get that 𝑏 is equal to 9.509 centimeters because it’s the side length, and we rounded three decimal places after we plugged this in our calculator.
So now we’re left with finding lowercase 𝑐 and uppercase 𝐶, and we can’t solve for these, having two missing variables in one fraction. However, we can find capital 𝐶 right away. Capital 𝐶 is our missing angle, and all triangles add to 180 degrees. So angle 𝑐 plus 41 degrees plus 82 degrees should add to 180, so we can add 41 plus 82 and then subtract it from 180 and find that angle 𝑐 is equal to 57 degrees. So we can go ahead and replace capital 𝐶 with 57 degrees.
Now we can go ahead and solve for lowercase 𝑐. Just in case we did something wrong when we were solving for 𝑏, let’s go ahead and use everything that went with 𝑎, the 6.3 divided by sin of 41 and setting it equal to 𝑐 divided by the sin of 57. It’s a good testing strategy to not use values in a multistep problem, to use values that we have found, because just in case we made a mistake, we don’t wanna use some value that is incorrect.
So now we need to find the cross-product. Let’s go ahead and cross-multiply. 6.3 times the sin of 57 is equal to 𝑐 times the sin of 41. So now let’s go ahead and divide by the sin of 41 to both sides. They cancel on the right, and then after plugging in our calculator and rounding three decimal places, 𝑐 is equal to 8.054 centimeters.
Therefore, all three missing pieces that we found were 𝑏 equals 9.509 centimeters, 𝑐 equals 8.054 centimeters, and the measure of angle 𝐶 is equal to 57 degrees.