Chapter 6 Additional Topics in Trigonometry

6.1 The Law of Sines Objectives:  Use Law of Sines to solve oblique triangles (AAS or ASA).  Use Law of Sines to solve oblique triangles (SSA).  Find areas of oblique triangles.  Use Law of Sines to model & solve real-life problems. 2

Oblique Triangles  Have no right angles.  Angles labeled with capital letters.  Sides labeled with lower- case letters (same letter as opposite angle). 3

Proof of Law of Sines  Let h be the altitude of the triangle.  Find sin A and sin B. 4

Proof continued….  Solve each equation for h.  Set h = h. 5

The Law of Sines 6

Example 1  For ABC, C = 102.3°, B = 28.7°, and b = 27.4 feet. Find the remaining angle and sides. 7

Example 2  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? 8

Possible Combinations for Law of Sines  Given a, b, A, and where h = b sin A:  We will look at each one individually. 9

A is Acute and a < h  No triangle formed - a can’t reach the base. 10

A is Acute and a = h  One triangle is formed. 11

A is Acute and a > b  One triangle is formed – a intersects the base at only one point. 12

A is Acute and h < a < b (Ambiguous Case)  Two unique triangles can be formed.  Side a can intersect the base at 2 points. 13

A is Obtuse and a ≤ b  No triangle is formed – a can’t reach the base. 14

A is Obtuse and a > b  One triangle is formed. 15

The Ambiguous Case  Given A, a, and b, we can find h h = b sin A  Is a > b ? If so, only 1 triangle.  Is h < a < b ? If so, then 2 triangles. 16

Example 3  For ABC, a = 22 inches, b = 12 inches, and A = 42°. Find the remaining side and angles. 17

Example 4  For ABC, a = 12 meters, b = 31 meters, and A = 20.5°. a. How many triangles can be formed? b. Find all remaining side(s) and angles. 18

Example 5  Show that there is no triangle for which a = 15, b = 25, and A = 85°. 19

Area of an Oblique Triangle  The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle.  What happens if the included angle is 90°? 20

Example 6  Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102°. 21

Homework 6.1  Worksheet