Section 6.1 Law of Sines

Objective By following instructions students will be able to: Use the Law of Sines to solve oblique triangles (AAS, ASA, or SSA). Find areas of oblique triangles. Use the Law of Sines to model and solve real-life problems.

Oblique Triangles C Triangles that have no right angles. When Solving, you will need: AAS or ASA SSA SSS SAS a b A B c

Law of Sines If triangle ABC is a triangle with sides a, b, and c, then: or

Example 1: Given Two Angles and One Side-AAS In triangle ABC, , , and . Find the remaining angle and side.

Example 2: Given Two Angles and One Side-AAS A pole tilts toward the sun at an 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 . How tall is the pole?

The Ambiguous Case – SSA If Angle A is ACUTE Consider a triangle in which you are given a, b, and A; (h = b sinA) One Triangle A b h a A b a h

The Ambiguous Case – SSA If Angle A is ACUTE Consider a triangle in which you are given a, b, and A; (h = b sinA) Two Triangles No Triangle A b a a h A b a h

The Ambiguous Case – SSA If Angle A is OBTUSE Consider a triangle in which you are given a, b, and A; (h = b sinA) No Triangle One Triangle A b a A b a

Example 3: Single Solution Case- SSA For the triangle, , , and . Find the remaining side, and angles. A b=12in a=22in C B c

Example 4: No Solution Case- SSA Show that there is no triangle for which, , , and . A b=25 a=15 h

Example 5: Two Solution Case- SSA Find two triangles for which , , and .

Area of an Oblique Triangle Consider a triangle in which you are given a, b, and A; (h = b sinA) The area of any triangle is . Hence, C C A b a A b a h h B c c B

Example 6: Finding the Area of an Oblique Triangle Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of . C b=52m a=90m

Example 7: An Application of the Law of Sines The course for a boat race starts at point A and proceeds in the direction South West to point B, then in the direction South East to point C, and finally back to A. The point C lies 8 kilometers directly south of point A. Approximate the total distance of the race course.

Revisit Objective Did we… Use the Law of Sines to solve oblique triangles (AAS, ASA, or SSA)? Find areas of oblique triangles? Use the Law of Sines to model and solve real-life problems?

Homework Pg 434 #1, 4, 5, 7, 8, 12-14, 19, 20