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After your quiz… Solve for all missing angles and sides: x 3 5 Y Z

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

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

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LG 4-2 Law of Sines and Law of Cosines MA3A6. Students will solve trigonometric equations both graphically and algebraically. d. Apply the law of sines and the law of cosines.

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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. A B C a b c A C B a b c

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What we already know The interior angles total 180. We cant 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. A B C a b c

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What is the Law of Sines? A B C a b c Drop an altitude and call it h. h h is opposite to both A and B, then Solve both for h. This means

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A B C a b c If we drop an altitude to side a, you get: Putting it all together gives us the Law of Sines: (You can also use it upside-down)

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Use Law of SINES when... AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side SSA – (SOMETIMES) 2 sides and their adjacent angle …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 wont have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given:

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General Process for Law Of Sines 1.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. 2.Once you know 2 angles, you can subtract from 180 to find the 3 rd. 3.To avoid rounding error, use given data instead of computed data whenever possible.

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

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

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Example 3: Solve this triangle A B C a b c 3510 45 Since we cant start one of the fractions, well start by finding C. C = 180 – 35 – 10 = 135 135 Since the angles were exact, this isnt a rounded value. We use sinC/c as our starting fraction. Using your calculator 36.5 11.1

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You try! Solve this triangle AC B 115° 30° a = 30 c b

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

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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 General Strategies for Using the Law of Cosines

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87.0° 15.017.0 c B A Example 1: Solve this triangle Use the relationship: c 2 = a 2 + b 2 – 2ab cos C c 2 = 15 2 + 17 2 – 2(15)(17)cos(87°) c 2 = 487.309… c = 22.1 Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines.

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

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You TRY: 1.Solve a triangle with a = 8, b =10, and c = 12. 2.Solve a triangle with A = 88 o, B =16 o, and c = 14. A = 41.4 o a = 8 B = 55.8 o b = 10 C = 82.8 o c = 12 A = 88 o a = 12.4 B = 16 o b = 3.4 C = 76 o c = 14

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

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Triangle Inequalities

Triangle Inequalities

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