7-6 The Law of Sines

Transcription

1 7-6 The Law of Sines So far, we have learned how to use geometric mean, Pythagorean Theorem, properties of and triangles, and Soh, Cah, Toa to solve triangles. The Law of Sines is used to solve triangles that are not triangles (It can be used to solve right triangles also). Consider ABC with altitude CD. Use h to represent the measure of the altitude. 1) Identify the right triangles. C 2) Find sin A and sin B. A b h D a B 3) Solve for h. 4) Substitute. 5) Rewrite as a proportion. Say hello to the LAW OF SINES!.

2 Law of Sines Let ABC be any triangle with a,b, and c representing the measures of sides opposite angles with measures A, B, and C respectively. Then, = = Refer to the figure to complete each statement.

3 To "solve a triangle" means to find all of the missing sides and angles m X = x = z =

4 Using Law of Sines sina sinb sinc = a b c = B 7 85 o a Solve ABC A b 38 o C m m m A = B= C= a= b= c= 85 38

5 Applying to word problems: To find the distance between two points, A and C, that are on opposite sides of a river, a surveyor measures the distance to point B, on the same side of the river as point A. The distance from A to B is 240 feet. He then measures the angle from A to C as 62 o and the measure of the angle from B to C is 55 o. Find the distance from A to C. A B C

6 Use the Law of Sines Example 1: Find p. Round to the nearest tenth. 8 Q P 17º 28º R Example 2: Find the measure of angle L to the nearest degree in triangle LMN if n = 7, l = 9, and the measure of angle N = 43. Remember: To solve a triangle means to find the measures of ALL the angles and sides of a triangle. Example 3: Solve DEF if m D = 112, m F = 8, and f = 2.

7 Recall: I can use: If I know two sides of a right triangle and need to find the third side, If I know two sides of a right triangle and need to find an angle measure, I can use: If I know one side and one angle of a right triangle and need to find another side, I can use: If I know two sides and one opposite angle of a non- right triangle and need to find the other opposite angle, I can use: If I know two angles and one opposite side of a non- right triangle and need to find the other opposite side, I can use: What if we know all three sides of a non- right triangle and need to find the measures of the angles?

8 Law of Cosines Warm-ups Solve for x. A. 12 = 6 2x B. 2 = x C. 13 = cosx D. a 2 = (4)(7)cos 52 E = (8.8)(12.1)cos T

9 7-7 Law of Cosines The Law of Cosines, like the Law of Sines, is used to solve triangles that are not necessarily right. The Law of Cosines, however, allows us to solve triangles that cannot be solved using Law of Sines. The formula for the Law of Cosines looks a little intimidating at first, but see if you can figure out the pattern and apply it to triangles whose names are not ABC. Ok, you try it: 2 For XYZ, y =

10 . g. 75 o g 3. Find x. X 45º 25 Y 11 x Z 4. A triangular field is fenced on two sides. The third side of the field is formed by a river. If the fences measure 150 meters and 98 meters and the side along the river is 172 meters, what is the measure of the angle between the fences?

11 5. Solve triangle ABC. Find the largest angle first!! Example: a = 12, b = 7, c = 9 12 B 9 C 7 A You never have to use Law of Cosines more than once when you're solving a triangle!! 6. EXTEND the problem: Rewrite the Law of Cosines into an alternate form that allows you to solve for angle B in one step!

12 For 7 and 8, sketch the triangle described then determine whether the Law of Sines or Law Cosines should be used first to solve the triangle. Recall: If I know two sides of a right triangle and need to find the third side, I can use: If I know two sides of a right triangle and need to find an angle measure, I can use: If I know one side and one angle of a right triangle and need to find another side, I can use: If I know two sides and one opposite angle of a non-right triangle and need to find the other opposite angle, I can use: If I know two angles and one opposite side of a non-right triangle and need to find the other opposite side, I can use: What if we know all three sides of a non-right triangle and need to find the measures of the angles?

13 Use Law of Sines when you know: 1) The measures of two angles and any side of the triangle. 2) The measure of two sides and an angle opposite one of those sides. Use Law of Cosines when you know: 1) The measure of three sides. 2) The measure of two sides and the included angle.

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