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-rightz triangle. 5 3

35o y x There are formulas to solve oblique triangles just like there are for right triangles! Lesson 4-7 Solving Oblique Triangles Laws of Sines and Cosines

MM4A6. Students will solve trigonometric equations both graphically and algebraically. d. Apply the law of sines and the law of cosines.

Introductory Comments C You have learned to solve right triangles in previous sections. b a Now we will solve oblique triangles (non-right triangles). A c B

Note: Angles are Capital letters C and the side opposite is the a same letter in lower case. b

A c B Introductory Comments • The interior angles total 180. • We can’t use the Pythagorean C Theorem. Why not? b a • Larger angles are across from longer sides and vice versa. A c B • The sum of two smaller sides must be greater than the third.

The Law of Sines helps you solve for sides or angles in an oblique triangle.

sinABC sin sin = = abc

(You can also use it upside-down) abc = = sinABC sin sin Use Law of SINES (LOS) 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 LOS 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 ASA and SSS triangles, 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 angle 3. To avoid rounding error, use given data instead of computed data whenever possible. Example 1 Solve this triangle: B

80° 12 c

A 70° C b Example 2: Solve this triangle C

b a =30 50 45 B A c Example 3: Solve this triangleC Since we can’t start one of the fractions, we’ll start by finding C. 135 11.1 b a 36.5 C = 180 – 35 – 10 = 135 35 10 B Since the angles were exact, this A c 45 isn’t a rounded value. We use sin135 sin 35 sin135 sin10 = = sinC/c as our starting fraction. 45 a 45 b sinCA sin sin CB sin = and = ca cb a sin135= 45sin 35 bsin135= 45sin10 45sin 35 45sin10 a = b = sin135 sin135 Using your calculator a ≈ 36.5 b ≈ 11.1 You try! Solve this triangle

B

30° c a = 30

115° C A b So, what about this one?

B

30° C = 42 a = 30

C A b 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 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:

a2=+− b 22 c 2bc cos(A) b2=+− a 22 c 2ac cos(B) c2=+− a 22 b 2ab cos(C) Example 1: Solve this triangle

87.0° 17.0 15.0

A B c Example 2: Solve this triangle C

31.4 23.2

α β

38.6 You TRY: 1. Solve a triangle with a = 8, b =10, and c = 12. Hint: find the largest angle first, then the others. 2. Solve a triangle with a = 16, B = 63o, and c = 12.

IMPORTANT

• Always try the LOS first. If you can’t use it (SSS or SAS), use LOC to find one missing measure and revert back to LOS to finish. • If you need to find an angle using LOC, find the largest angle first, then the others.