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7.1 The Sines Law

Determining if the Law of Sines Can Be Used to Solve an Oblique Triangle Using the Law of Sines to Solve the SAA Case or ASA Case Using the Law of Sines to Solve the SSA Case Using the Law of Sines to Solve Applied Problems Involving Oblique Triangles. 7.1 The Sines Law.

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7.1 The Sines Law

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  1. Determining if the Law of Sines Can Be Used to Solve an Oblique Triangle • Using the Law of Sines to Solve the SAA Case or ASA Case • Using the Law of Sines to Solve the SSA Case • Using the Law of Sines to Solve Applied • Problems Involving Oblique Triangles 7.1 The Sines Law Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu

  2. The Law of Sines If A, B, and C are the measures of the angles of any triangle and if a, b, and c are the lengths of the sides opposite the corresponding angles, then

  3. The Law of Sines The following information is needed to use the Law of Sines. 1. The measure of an angle must be known. 2. The length of the side opposite the known angle must be known. 3. At least one more side or one more angle must be known.

  4. Determining of the Law of Sines Can Be Used to Solve an Oblique Triangle Decide whether or not the Law of Sines can be used to solve each triangle. a. b. c. Yes; we are given an angle and the measure of its opposite side and an additional angle. Yes; we are given an angle and the measure of its opposite side and an additional angle. No; We are not given the measure of any angle.

  5. C The sum of the measurements of a triangle’s interior angles is 180º. A+B+C= 180º 50º +B+ 33.5º = 180º A = 50º and C = 33.5º. 33.5º a b =76 83.5º +B= 180º Add. B= 96.5º Subtract 83.5º from both sides. 50º A B c Text Example (ASA) Solution We begin by drawing a picture of triangle ABC and labeling it with the given information. The figure shows the triangle that we must solve. We begin by finding B. • Solve triangle ABC if A= 50º, C= 33.5º, and b= 76.

  6. C This is the known ratio. 33.5º b =76 a 50º A B c Text Example cont. Solution Keep in mind that we must be given one of the three ratios to apply the Law of Sines. In this example, we are given that b= 76 and we found that B= 96.5º. Thus, we use the ratio b/sin B, or 76/sin96.5º, to find the other two sides. Use the Law of Sines to find a and c. Solve triangle ABC if A= 50º, C= 33.5º, and b= 76. Find a: Find c: The solution is B= 96.5º, a 59, and c 42.

  7. Solving a SAA Triangle Using the Law of Sines Solve the given oblique triangle. Round lengths to one decimal place.

  8. Solving a SAA Triangle Using the Law of Sines Solve the given oblique triangle. Round lengths to one decimal place. Find a: Find b:

  9. Solving a ASA Triangle Using the Law of Sines Solve oblique triangle ABC if B = 42, A = 57, and c = 18.6 cm.

  10. a=h andisjust theright length to form a right triangle. a is less than h and not long enough to form a triangle. a b b h=bsinA h=bsinA a A A a is greater than h and a is less than b. Two distinct triangles are formed. a is greater than h and a is greater than b. One triangle is formed. b a b h=bsinA a a A A The Ambiguous Case (SSA) No Triangle One Right Triangle Two Triangles One Triangle

  11. Solving a SSA Triangle Using the Law of Sines ( Quiz No) Solve the triangle; if possible. Since there is no angle A for which sin A > 1, there can be no triangle with the given measurements.

  12. Solving a SSA Triangle Using the Law of Sines (One Triangle) Solve the triangle; if possible. Two possible angles:

  13. Solving a SSA Triangle Using the Law of Sines (One Triangle)--cont Solve the triangle; if possible.

  14. Solving a SSA Triangle Using the Law of Sines (Two Triangles) Solve the triangle; if possible. Two possible angles: There are two possible triangles.

  15. Solving a SSA Triangle Using the Law of Sines (Two Triangles) Solve the triangle; if possible.

  16. Solving a SSA Triangle Using the Law of Sines (Two Triangles) Solve the triangle; if possible.

  17. Solving an Applied Problem Martin wants to measure the distance across a river. He has made a sketch. Find the distance across the river, a. a

  18. Determining the Distance a Ship is from Port A ship set sail from port at a bearing of N 53E and sailed 63 km to point B. The ship then turned and sailed an additional 69 km to point C. Determine the distance from port to point C if the ship’s final bearing is N 74E. Draw a diagram. Find angle A.

  19. Determining the Distance a Ship is from Port-cont A ship set sail from port at a bearing of N 53E and sailed 63 km to point B. The ship then turned and sailed an additional 69 km to point C. Determine the distance from port to point C if the ship’s final bearing is N 74E. Two possible angles:

  20. Determining the Distance a Ship is from Port-cont A ship set sail from port at a bearing of N 53E and sailed 63 km to point B. The ship then turned and sailed an additional 69 km to point C. Determine the distance from port to point C if the ship’s final bearing is N 74E. The ship is 124 miles from port to point C.

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