Section 1.3.1

Law of Sines and Area

SAS Area1-90 Since SAS determines a triangle, it should be possible to find its area given the 2 sides and the angle between those sides. Since the letter A is already used as a vertex of our generic triangle, use the letter K to denote the area of the triangle.

x

5 8

4

y

8

B

A

C

a) Write an expression for the area, K, of triangle ABC.

b) Using angle C and hypotenuse b, find an expression for the height h.

c) Now combine your results from parts a and b to find a formula for the area in terms of angle C and sides a and b.

1-91 Two adjacents sides of a triangle are 4 cm and 6 cm in length, the angle between them is 76 degrees. Use the formula found in part c of the last problem (called SAS formula to find the area of the triangle.

1sin

2k ab C

• We just derived (proved) a formula for the area K of ΔABC in terms of side b, side c, and angle A of an acute triangle. We did not consider what happens when the triangle is obtuse. Use the diagram below to prove the formula above works for an obtuse triangle.

• Tomas, an eccentric man who loves triangular shapes, was in the process of building his dream home. Of course, it was to be built out of triangles. In this case, Tomas was building a workshop with a square floor and triangular sides of length 10 feet from the base to the top, with apex angles of 40°. Find the surface area of the sides of his workshop.

Law of Sines: ASA and AAS Triangles

1-94 Use the diagram to complete the following problems, given triangle ABC is acute.

x

5 8

4

y

8

BA

Ca) Write an expression to express h in terms of angle A and side b.

b) Write an equation to express h in terms of angle B and side a.

c) Use your results from parts a and b to show that sin sin.

A Ba b

Law of Sines: ASA and AAS Triangles

1-94 Use the diagram to complete the following problems, given triangle ABC is acute.

x

5 8

4

y

8

BA

Cd)

e)

A B

C

a

c

bsin sin sina b cA B C

The Law of Sines is:

sin sin sinA B Ca b c

or

Example:

CT

A

8 cm

Calculator and Trigonometry

ROUNDING in the AP classes

Degrees, Minutes, Seconds

p. 36

Examples: Convert 16o 24’ 18” to Degrees, Decimals

Convert 67.35827o to Degrees ,Minutes, Seconds

Assignment

Pg 37  #1-92, 1-93, 1-96 TO 1-101