Using the Law of Sines and Cosines to Find Angles 04:08 minutes

Video Transcript

Transcript Using the Law of Sines and Cosines to Find Angles

Pete manages lighthouses on three islands: Aurora's Alluvia, Bruce's Bluff and Clark's Cliff. Pete, the lighthouse keeper, notices that one of the three lighthouses he manages has just gone out. Pete must act quickly before disaster strikes, but it's slow going because the fog is as thick as split pea soup. How will he find the lighthouse in all this fog? No worries. Pete can use the Law of Sines and Cosines to Solve for Unknown Angles.

Law of Cosines

To calculate the angle of a non-right triangle when you know all the sides, or just two sides and one angle, you can use the Law of Cosines. Since Pete knows the distances between the three lighthouses. He just needs to figure out the unknown angle, and then set the course of his boat.
Let’s review the formulas. To make these formulas easier to memorize, notice the pattern - it's similar to the Pythagorean Theorem with some extras. Let's take a look. Does the first part of the equation look familiar? It should it's the Pythagorean Theorem! Notice a pattern? That's right. All we have to do to get the Law of Cosines is subtract the product of the cosine of the angle opposite the side you're trying to find and 2 times the product of the remaining sides from our original Pythagorean Theorem.

But which formula should we use? Since the unknown quantity is ∠A, we can use this formula, but first, let's modify it to solve for cos(A). Oh boy, that looks really complicated - we may need a calculator for this. Let’s try it out. Pete already knows that the distance between Bruce's Bluff and Clark's Cliff is equal to 5.18 miles, between Aurora's Alluvia and Clark's Cliff is 9 miles and the distance between Aurora's Alluvia and Bruce's Bluff is 6 miles. Pete plugs in the known distances into the Law of Cosines formula. Ok now that Pete knows the angle he can set off for Clark's Cliff.
Oh no… the replacement bulb is overboard! To get a replacement bulb he'll have to go to Bruce's Bluff. But now he'll need to chart a course to get from Bruce's Bluff to Clark's Cliff.

Law of Sines

He can calculate the cosine of angle B, but since he knows at least two sides and one of the opposite angles now, he can use the Law of Sines. Take a look at the formula for the Law of Sines. The lower case letters indicate the length of the three sides, and the upper case letters indicate the measure of the angles opposite the respective sides. Look! You can also turn the formula upside down. Use the formula that's easiest for you to remember to solve for the unknown measurement.

Okay, let’s use what we know to solve for angle B. Since Pete knows the distances to and from each of the islands as well as the degree measure between AB and AC, he can use the Law of Sines to find the direction of Clark's Cliff. Pete plugs in the known distances, this time, into the Law of Sines formula. Ok...Pete's all set to fix the light bulb.

But through the fog, he gets distracted by a mermaid? Pete decides a LITTLE detour might not be the worst thing. Oops, that’s not a mermaid…