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After your quiz… Solve for all missing angles and sides: x 3 5 Y Z
What formulas did you use to solve the triangle? Pythagorean theorem SOHCAHTOA All angles add up to 180 o in a triangle
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!
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.
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
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
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
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)
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:
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.
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
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.
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
You try! Solve this triangle AC B 115° 30° a = 30 c b
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
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
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.
Example 2: Solve this triangle 31.4 23.2 38.6 C We start by finding cos A.
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
Practice Do multiples of 3 in class. Turn in your answer to #6 and #15 Complete the rest for HW