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Solving Oblique Triangles Using Sine and Cosine Laws in Trigonometry

This lesson covers the essential techniques for solving oblique triangles through the Sine and Cosine Laws. We will learn how to calculate angles and lengths of oblique triangles when given specific side lengths and angle measurements. Key concepts include understanding the relationship between sides and angles, applying the Law of Sines and Cosines, and deriving the area of triangles. Examples will illustrate the use of trigonometric functions (sine, cosine, tangent) and how to identify when each law should be applied for efficient problem-solving.

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Solving Oblique Triangles Using Sine and Cosine Laws in Trigonometry

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  1. Trigonometry

  2. Unit 4:Mathematics Aims • Solve oblique triangles using sin & cos laws Objectives • Calculate angles and lengths of oblique triangles.

  3. B c=5.2 a=2.4 A b=3.5 C

  4. The table shows some of the values of these functions for various angles.

  5. Sines increase from 0 to 1 Between 0o a 90o:

  6. Cosines decrease from 1 to 0 Between 0o a 90o:

  7. Tangents increase from 0 to infinity.

  8. Cos(90 - X) = Sin(X)Sin(90 - X) = Cos(X)

  9. 1. 45º 6. 63º 7. 90º 2. 38º 8. 152º 3. 22º 9. 112º 4. 18º 10. 58º 5. 95º Write out the each of the trigonometric functions (sin, cos, and tan) of the following

  10. B c a A C b When solving oblique triangles, simply using trigonometric functions is not enough. You need… The Law of Sines The Law of Cosines a2=b2+c2-2bc cosA b2=a2+c2-2ac cosB c2=a2+b2-2ab cosC

  11. REMEMBER Whenever possible, the law of sines should be used. Remember that at least one angle measurement must be given in order to use the law of sines. The law of cosines in much more difficult and time consuming method than the law of sines and is harder to memorize. This law, however, is the only way to solve a triangle in which all sides but no angles are given. Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.

  12. The triangle has three sides, a, b, and c. There are three angles, A, B, C (where angle A is opposite side a, etc). The height of the triangle is h. The sum of the three angles is always 180o. A + B + C = 180o

  13. The area of this triangle is given by one of the following three formulae Area = (a × b × Sin C) = (a × c × Sin B) = 2 2 (b × c × Sin A) 2 = b × h 2

  14. The relationship between the three sides of a • general triangle is given by • The Cosine Rule. • There are three forms of this rule. All are equivalent. a2 = b2 + c2 - (2 × b × c × Cos A) b2 = a2 + c2 - (2 × a × c × Cos B) c2 = a2 + b2 - (2 × a × b × Cos C)

  15. Show that Pythagoras' Theorem is a special case of the Cosine Rule. In the first version of the Cosine Rule, if angle A is a right angle, Cos 90o = 0. The equation then reduces to Pythagoras' Theorem. a2 = b2 + c2 - (2 × b × c × Cos 90o) = b2 + c2 - 0 = b2 + c2 The relationship between the sides and angles of a general triangle is given by The Sine Rule.

  16. Find the missing length and the missing angles in the following triangle. By the Cosine Rule, a2 = b2 + c2 - (2 × b × c × Cos A)

  17. Find the missing length and the missing angles in the following triangle. Now, from the Sine Rule, This can be rearranged to

  18. REMEMBER Side a is opposite angle A Side b is opposite angle B Side c is opposite angle C

  19. B B B B c c 22 25 31 º 12 5 15 a 168 º 35 º 28 º A A A A 24 8 14 b C C C C Solve the following oblique triangles with the dimensions given

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