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Understanding the Law of Sines and Cosines in Triangles

In this chapter on trigonometry, we explore the Law of Sines and its application in finding unknown measurements in triangles. We demonstrate how any triangle can be divided into two right triangles using an altitude. By applying the Law of Sines, we solve for unknown sides and angles using various cases such as ASA and AAS. Additionally, the Law of Cosines is introduced for situations where the Law of Sines may not apply, particularly for SAS and SSS configurations. Assignments are included to reinforce understanding of these concepts.

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Understanding the Law of Sines and Cosines in Triangles

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Presentation Transcript

  1. Analytic Trigonometry Barnett Ziegler Bylean

  2. Additional Triangle Ratios Chapter 6

  3. Law of sines Ch6 - section 1

  4. Deriving the law of sines • Any triangle can be split into 2 right triangles by dropping an altitude from one vertex to the base opposite • sin( • With a little algebra • By rotating the triangle to a different base you can also get b a h

  5. Using the law of sines to find measurements • Given an angle , a side, and a second angle (the side between the angles) (ASA) • Find the 2nd and 3rd sides • The missing angle that is opposite the known side is 105⁰ 45⁰ 30⁰ 55in

  6. By law of sines • Solving each separately • and

  7. Another example • Similarly given 2 angles and the side Not between them (AAS) • By law of sines 42ft 80 40

  8. Angle side side • 3 possible answers • No triangle – 35 , 12 in, 4 in • One triangle 40 , 4.8 units, 5.5 units • Two triangles 40, 6units, 5 units

  9. Assignment • p361(5-50 odd, 57-78 odd)

  10. Law of cosines Chapter 6 – section 2

  11. Given side angle side (SAS) • Law of sines won’t work – • given 5 in 42 and 8 in

  12. Law of cosines • given 5 in 42 and 8 in • Now you can use law of sines to find the other measures

  13. Given side sideside • Again law of sines is useless • 8 in, 12 in, 17 in • First verify that you have a triangle a + b > c (triangle inequality)

  14. assignment • P372(5-42odd,51-69)

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