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Lawsof Cosines - Example 1

In mathematics, the law of cosines, also known as the cosine rule or the cosine formula, is a formula used to calculate the cosine of an angle in a triangle, given the lengths of the sides of the triangle and the angle itself. It is one of the basic facts in Euclidean geometry.

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Co signs to solve the sides and angles of triangle f g H were given That angle G is 82 degrees side elf. It's six degrees and side H is four degrees. So since we're given those rules, we can go ahead and we can use that with the law of coastlines. So we're giving side elf were given side H and were given angle G. So we're need to find side G first. So soggy we're gonna lol Kasan says that G squared equals F squared plus h squared minus two f h Kasan of Angle G were given all that information so we can plug that in so we know G squared F squared. We've already said that fs six. So we're gonna have six squared plus ages four. So we'll have four squared minus two times six Tom's four cosign and were given angle G and Angle G is 82 degrees, so there's all of our values plugged in, so let's go ahead and simplify this some. So six squared is 36 4 square to 16. If I multiply two times six times four, we get 48 cosign of 82 degrees. Simplify this even more 36 16 will give me 52 minus 48 times. Cosign of 82 is approximately 6.680 So that would leave me at G. Squared is approximately 45.3. So now that I know that G square approximately US 45.3, we need to find out what G is approximately So we need to find the square root of G squared, which is G and the square root of 45.3. And the square root of 45.3 is approximately 6.7. So Side G is approximately 6.7. So now that we know that we can go ahead and we can find our other two angles and now we're gonna use the lawless signs So we're no, we've been given angle G and we've already found Side G. So let's go ahead and let's start with angle. If so, a line of sign says that sign of G divided by side G equal sign of elf divided by side F. So sign of G. We said G is 82 so sign of G, the sound of 82 degrees over Weed said Soggy is 6.7. Sign of elf. We don't know. That's what we're looking for Divided by We've said that side of elf iss six So now we're gonna cross multiply So we have 67 times Sign of elf would equal six times son of 82 degrees. Since we're looking for a sign of elf I'm gonna go ahead and set this up where we're gonna divide by 6.7 So we have son of Elf will equal six times Sign of 82 degrees is approximately 5.942 and we already know 6.74 side elf. When I divide that, I'm gonna get in approximate value of 0.887 now to find angle If we need to do the reverse operation in our calculator which would be our second sign button, we would type the sin. So that would tell me that angle elf is approximately 62 degrees. So ankle elf is 62 degrees. So now that we know angle f and angle G finding angle H is just using our rules of angles in a triangle. We know that angle F plus Angle G plus ankle H altogether is gonna equal 180 well, f. We've already said it's 62 degrees G is 82 degrees and 62 plus 82 is 144 so I can subtract 1 80 minus 1 44 and get angle age. And so that means that angle age is going to be 36 degrees.