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Warm UP! Solve for all missing angles and sides: Z 5 3 Y x What formulas did you use to solve the triangle? • Pythagorean theorem • SOHCAHTOA • All angles add up to 180o in a triangle Could you use those formulas on this triangle? Solve for all missing angles and sides: This is an oblique triangle. An oblique triangle is any non-rightztriangle. 5 3 y 35o x There are formulas to solve oblique triangles just like there are for right triangles! Solving Oblique Triangles Laws of Sines and Cosines & Triangle Area Students will solve trigonometric equations both graphically and algebraically. . Apply the law of sines and the law of cosines. General Comments C You have learned to solve right triangles . 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 A B c C a b A c B 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. C a b A c B The Law of Sines helps you solve for sides or angles in an oblique triangle. sin A sin B sin C a b c (You can also use it upside-down) a b c sin A sin B sin C 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 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 3rd. 3. To avoid rounding error, use given data instead of computed data whenever possible. Example 1 Solve this triangle: The angles in a ∆ total 180°, so solve for angle C. B 80° c A Set up the Law of Sines to find side b: 12 70° C b 12 b sin 70 sin 80 12 c sin 70 sin 30 12 sin 80 b sin 70 12 sin 30 c sin 70 12 sin 80 b 12.6cm sin 70 c 12 sin 30 6.4cm sin 70 Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Example 2: Solve this triangle C 85 b a =30 50 45 A c sin A sin C a c sin 45 sin 85 30 c c sin 45 30sin85 30sin85 c sin 45 You’re given both pieces for sinA/a and part of sinB/b, so we start there. sin 45 sin 50 B 30 b b sin 45 30sin 50 30sin 50 b sin 45 Using a calculator, b 32.5 Using a calculator c 42.3 Example 3: Solve this triangle C Since we can’t start one of the fractions, we’ll start by finding C. 11.1 C = 180 – 35 – 10 = 135 Since the angles were exact, this isn’t a rounded value. We use sinC/c as our starting fraction. sin C sin A sin C sin B and c a c b b 135 a 35 A sin135 sin 35 45 a a sin135 45sin 35 a 45sin 35 sin135 10 c 45 B sin135 sin10 45 b b sin135 45sin10 b 45sin10 sin135 Using your calculator a 36.5 36.5 b 11.1 You try! Solve this triangle B 30° c a = 30 C 115° b A Example 3-Application A forest ranger at an observation point (A) sights a fire in the direction 32° east of north. Another ranger at a second observation point (B), 10 miles due east of A, sights the same fire 48° west of north. Find the distance from each observation point to the fire. 6.795 80o 48o 42o 32o 58o A 8.611 10 B Example A civil engineer wants to determine the distances from points A and B to an inaccessible point C, as A shown. From direct measurement, the engineer knows that AB = 25m, A = 110o, and B = 20o. Find AC and BC. C B 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: a b c 2bc cos(A) 2 2 2 b a c 2ac cos(B) 2 2 2 c a b 2ab cos(C) 2 2 2 General Strategies for Using the 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 Example 1: Solve this triangle 87.0° 17.0 15.0 B A Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines. c sin 87.0 sin A 22.1 15.0 Use the relationship: c2 = a2 + b2 – 2ab cos C c2 = 152 + 172 – 2(15)(17)cos(87°) c2 = 487.309… c = 22.1 42.7 sin 87.0 sin B 22.1 17.0 50.2 Example 2: Solve this triangle C sin B sin 36.9 31.4 23.2 23.2 31.4 38.6 We start by finding cos A. a 2 b 2 c 2 2bc cos A sin C sin 36.9 38.6 23.2 cos A 0.7993 A 36.9 54.4 C 87.3 You TRY: 1. Solve a triangle with a = 8, b =10, and c = 12. A = 41.4o B = 55.8o C = 82.8o a= 8 b = 10 c = 12 2. Solve a triangle with A = 88o, B =16o, and c = 14. A = 88o B = 16o C = 76o a = 12.4 b = 3.4 c = 14 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