Solving Triangles with Law of Sines

Learn how to solve oblique triangles using the Law of Sines, including the ambiguous case. Practice finding missing sides and angles in triangles without right angles. Explore examples and applications to enhance your trigonometry skills.

• Trigonometry
• Law of Sines
• Oblique Triangles
• Ambiguous Case
• Application Problems

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1. Warm Up Solve the following triangles for the missing side or angle: 1) 2) 3) 9 10 32 48 14 27 8 x x

2. Law of Sines Section 6.1

3. Objectives Students will be able to Solve oblique triangles using the Law of Sines (including the ambiguous case) Solve application problems for all triangles using trigonometry

4. Homework Law of Sines Worksheet - EVENS

5. Oblique Triangles Triangles without a right angle Four Cases: C 1. Two angles and any side (AAS or ASA) a b 2. Two sides and an angle opposite one of them (SSA) (ambiguous case) A B c 3. Three sides (SSS) 4. Two sides and their included angle (SAS) First 2 cases can be solved using the Law of Sines

6. Law Of Sines If ABC is a triangle with sides a, b, and c, then ? ????= ? ? ????= ???? C C a b a b 0 0 A c B B c A *Can be written as ???? =???? =???? ? ? ?

7. Example - AAS Given C = 102.3 , B = 28.7 and b = 27.4 feet, find the remaining angles and sides C a b A B c

8. Example - ASA A pole tilts toward the sun at an 8 angle from the vertical and it casts a 22 foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43 . How tall is the pole? 0

9. Ambiguous Case - SSA In a triangle where you are given a, b and h (h = b sinA) If angle A is acute And a < h no triangle is possible And a = h one triangle is possible (right triangle) And a b one triangle is possible And h < a < b two triangles are possible If angle A is obtuse And a b no triangle is possible And a > b one triangle is possible

10. Example Given a = 15, b = 25, and A = 85 , solve the triangle.

11. Example Given a = 22 in, b = 12 in, and A = 42 , find the remaining side and angles.

12. Example Solve the triangle, given a = 12 m, b = 31 m, and A = 20.5 .

14. Example A bridge is to be built across a small lake from a gazebo to the dock is S 41 W. From a tree 100m from the gazebo, the bearings to the gazebo and the dock are S 74 E and S 28 E respectively. Find the distance from the gazebo to the dock. T G D

15. Practice Law of Sines Worksheet

16. Closure Given A = 102.4 , C = 16.7 , and a = 21.6, solve for the remaining angle and sides