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The Law of Cosines

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  1. The Law of Cosines

  2. Let's consider types of triangles with the three pieces of information shown below. We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. SAS AAA You may have a side, an angle, and then another side You may have all three angles. AAA This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down" SSS You may have all three sides

  3. Let's place a triangle on the rectangular coordinate system. (a cos , a sin ) (x, y) a c y  x b (b, 0) This is the Law of Cosines

  4. LAW OF COSINES LAW OF COSINES We could do the same thing if gamma was obtuse and we could repeat this process for each of the other sides. We end up with the following: Use these to findmissing sides Use these to find missing angles

  5. Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA). Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA).

  6. Solve a triangle where b = 1, c = 3 and  = 80° Draw a picture. This isSAS  3 a Do we know an angle and side opposite it? No so we must use Law of Cosines.  80 1 Hint: we will be solving for the side opposite the angle we know. minus 2 times the productof those other sides times the cosine of the angle between those sides One side squared sum of each of the other sides squared CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction a = 2.99

  7. We'll label side a with the value we found. We now have all of the sides but how can we find an angle?  3 19.2 2.99 80.8  80 Hint: We have an angle and a side opposite it. 1  is easy to find since the sum of the angles is a triangle is 180° NOTE: These answers are correct to 2 decimal places for sides and 1 for angles. They may differ with book slightly due to rounding. Keep the answer for in your calculator and use that for better accuracy.

  8. Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture. This isSSS 9  5 Do we know an angle and side opposite it? No, so we must use Law of Cosines. 84.3   8 Let's use largest side to find largest angle first. minus 2 times the productof those other sides times the cosine of the angle between those sides One side squared sum of each of the other sides squared

  9. How can we find one of the remaining angles? Do we know an angle and side opposite it?  9 62.2 5 84.3   33.5 8 Yes, so use Law of Sines.

  10. Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Example: 10 Find the area of the triangle. 8 5