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Warm UP!

Z. Warm UP!. 5. 3. Y. Solve for all missing angles and sides:. x. What formulas did you use to solve the triangle?. Pythagorean theorem SOHCAHTOA All angles add up to 180 o in a triangle. This is an oblique triangle. An oblique triangle is any non-right triangle.

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Warm UP!

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  1. Z Warm UP! 5 3 Y Solve for all missing angles and sides: x

  2. What formulas did you use to solve the triangle? • Pythagorean theorem • SOHCAHTOA • All angles add up to 180o in a triangle

  3. This is an oblique triangle. An oblique triangle is any non-right triangle. Could you use those formulas on this triangle? z 5 3 Solve for all missing angles and sides: y 35o x There are formulas to solve oblique triangles just like there are for right triangles!

  4. LG 5-2 Solving Oblique Triangles Laws of Sines and Cosines& Triangle Area MM4A6. Students will solve trigonometric equations both graphically and algebraically. d. Apply the law of sines and the law of cosines.

  5. LG 5-2 Solving Oblique Triangles • Laws of Sines and Cosines • Oblique Triangle Area • Practice 5-2/Review for Test • LG 5-2 TEST (1/31 ODD & 2/1 EVEN)

  6. C a b B A c General Comments You have learned to solve right triangles in ACC Math 2. Now we will solve oblique triangles (non-right triangles). Note: Angles are Capital letters and the side opposite is the same letter in lower case. C a b A c B

  7. C a b B A c What we already know • The interior angles total 180. • We can’t use the Pythagorean Theorem. Why not? • For later, area = ½ bh • Larger angles are across from longer sides and vice versa. • The sum of two smaller sides must be greater than the third.

  8. The Law of Sines helps you solve for sides or angles in an oblique triangle. (You can also use it upside-down)

  9. Use Law of SINES when ... …you have 3 parts of a triangle and you need to find the other 3 parts. They cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given: • AAS - 2 angles and 1 adjacent side • ASA - 2 angles and their included side • ASS– (SOMETIMES) 2 sides and their adjacent angle

  10. General Process for Law Of Sines • Except for the ASA triangle, you will always have enough information for 1 full fraction and half of another. Start with that to find a fourth piece of data. • Once you know 2 angles, you can subtract from 180 to find the 3rd. • To avoid rounding error, use given data instead of computed data whenever possible.

  11. B 80° 12 c 70° A C b Example 1 The angles in a ∆ total 180°, so solve for angle C. Set up the Law of Sines to find side b: Solve this triangle: Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm

  12. C 85 a b =30 50 B 45 A c Example 2: Solve this triangle You’re given both pieces for sinA/a and part of sinB/b, so we start there.

  13. C a b B A c Example 3: Solve this triangle Since we can’t start one of the fractions, we’ll start by finding C. C = 180 – 35 – 10 = 135 135 36.5 11.1 35 10 45 Since the angles were exact, this isn’t a rounded value. We use sinC/c as our starting fraction. Using your calculator

  14. You try! Solve this triangle B 30° c a = 30 115° C A b

  15. The Law of Cosines When solving an oblique triangle, using one of three available equations utilizing the cosine of an angle is handy. The equations are as follows:

  16. General Strategies for Usingthe Law of Cosines • The formula for the Law of Cosines makes use of three sides and the angle opposite one of those sides. We can use the Law of Cosines: • SAS - two sides and the included angle • SSS - all three sides

  17. Example 1: Solve this triangle Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines. 87.0° 17.0 15.0 B A Use the relationship: c2 = a2 + b2 – 2ab cos C c2 = 152 + 172 – 2(15)(17)cos(87°) c2 = 487.309… c = 22.1 c

  18. Example 2: Solve this triangle C 23.2 31.4 We start by finding cos A. 38.6

  19. You TRY: A = 41.4o a = 8 B = 55.8o b = 10 C = 82.8o c = 12 • Solve a triangle with a = 8, b =10, and c = 12. • Solve a triangle with A = 88o, B =16o, and c = 14. A = 88o a = 12.4 B = 16o b = 3.4 C = 76o c = 14

  20. IMPORTANT • IT IS ALWAYS BEST TO USE LAW OF SINES FOR SIDES AND LAW OF COSINES FOR ANGLES • Sometimes, however, it is just not possible – you may have to switch it up

  21. Practice • Do multiples of 3 in class. • Turn in your answer to #6 and #15 • Complete the rest for HW

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