review 1. solving a right triangle. 2. given two sides. 3. given one angle and one side
TRANSCRIPT
6.1: Law of Sines
Objectives:Use the Law of Sines to solve oblique trianglesFind areas of oblique trianglesUse Law of Sines to model & solve real-life problems
Oblique Triangles
Oblique triangles do not have right angles.
Triangles are usually labeled as:
A B
C
ab
c
Must haves for solving Oblique Triangles 2 angles and any side (AAS or ASA) 2 sides and an angle opposite one of
them (SSA) 3 sides (SSS) 2 sides and their included angle (SAS) The first 2 cases can be solved using
the Law of Sines The last 2 cases can be solved using
Law of Cosines
What is the Law of Sines?
Follow the directions on the Law of Sines Discovery notes ( available on mrtower.wordpress.com )
Law of Sines
If ABC is a triangle with sides a,b,c, then:
a
sin A b
sin B c
sinC
It can also be written as its reciprocal
sin A
asin B
bsinC
c
2 Angles & 1 Side (AAS)
Given: C=102.3º, B=28.7º, b=27.4 feet
Find: Finishing solving the triangle
A B
C
ab
c
1. Label the givens.
2. Solve for the missing angle.
3. Use the Law of Sines to find the 2 missing sides.
4. A = 49 degrees, a = 43.06 ft, c = 55.75 ft
2 Angles & 1 Side (ASA)Given: A pole tilts toward the sun at an 8º 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 43º.
Find: How tall is the pole?
AB
C
ab
c
Label the givens.Solve for the missing
angle.Use the Law of Sines
to find the 2 missing sides.
b = 23.84 ft, a = 34.62ft
The Ambiguous Case
am·big·u·ous /amˈbigyo; oəs/ Adjective
– (of language) Open to more than one interpretation; having a double meaning.
– Unclear or inexact because a choice between alternatives has not been made.
Synonyms– equivocal - vague - uncertain - doubtful - obscure
The Ambiguous Case (SSA)
This one is a pain… in the SSA.
Three possible situations:
1. No such triangle exists.
2. Only one such triangle exists.
3. Two distinct triangles can satisfy the conditions.
Example Show that there is no triangle for which
a=15, b=25, & A=85°
A
abh
1. Label the givens & draw picture.
2. Use the Law of Sines to find the missing angle B.
3. Is this result valid? Why or why not?
4. Invalid since out of Range
5. sinB = 1.66
ExampleGiven: triangle ABC where a=22 inches,
b=12 inches, & A=42°
Find: the remaining side and angles.
A B
C
ba
c
1. Label the givens & draw picture.
2. Use the Law of Sines to find the missing angle B.
3. Solve for C.
4. Solve for c.5. B=21o
C=117o c=29.29in
Example Find 2 triangles for which a=12 meters,
b=31 meters and A=20.5°1. Label the givens & draw both pictures.
2. Use the Law of Sines to find the missing angle B1.
3. Subtract B1 from 180° to find B2
4. Subtract the B and A values from 180° to find C1 and C2.
5. Use the Law of Sines to find c1 and c2.
6. Solution 1: B=64.8o C = 94.7o c = 34.15m
7. Solution 2: B=115.2o C=44.3o c= 23.93m
Area of an Oblique Triangle
Area 1
2bcsin A1
2absinC 1
2acsin B
The area of an oblique triangle given some angle is half the product of the two adjacent sides and the sine of
Example Find the area of a triangular lot having 2
sides of lengths 90 meters and 52 meters and an included angle of 102°
Label the givens & draw picture.Use the Area Formula to find the area of the lot.
Area = 2,288.82m2