6-1 LAW OF SINESChapter 6

OBJECTIVES

Use the law of sines to solve triangles

LAW OF SINES

For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

a

sin A

b

sin B

c

sin C

you have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given:

AAS - 2 angles and 1 adjacent side or ASA - 2 angles and their included side

SSS- three sides SSA (this is an ambiguous case)

EXAMPLE #1

You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c.

* In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .*

SOLUTION

The angles in a ∆ total 180°, so angle C = 30°.

Set up the Law of Sines to find side b:

A C

B

70°

80°a = 12

c

b

B:

12

sin 70

b

sin 8012 sin 80 b sin 70

b 12 sin80

sin 7012.6cm

12

sin 70

c

sin 3012 sin 30 csin70

c 12 sin 30

sin706.4cm

EXAMPLE#2

You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c.

SOLUTION

AC

B

115°

30°

a = 30

c

b

30

sin35

b

sin 30

30 sin 30 b sin 35

b 30 sin30

sin 3526.2cm

SOLUTION

30

sin 35

c

sin 11530 sin115 csin35

c 30 sin115

sin3547.4cm

CHECK IT OUT

For the triangle in fig 6.3 C=102.3 degree, B=28.7 degree and b=27.4 feet

C

A B

THE AMBIGUOUS CASE (SSA)

When given SSA (two sides and an angle that is NOT the included angle) , the situation is ambiguous. The dimensions may not form a triangle, or there may be 1 or 2 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist.

EXAMPLE

In the following examples, the given angle will always be angle A and the given sides will be sides a and b. If you are given a different set of variables, feel free to change them to simulate the steps provided here.

‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position

A B ?

b

C = ?

c = ?

AMBIGUOUS CASE

Situation I: Angle A is obtuse

If angle A is obtuse there are TWO possibilities If a ≤ b, then a is too short to reach side c - a

triangle with these dimensions is impossible.

A B ?

ab

C = ?

c = ?

If a > b, then there is ONE triangle with these dimensions.

A B ?

ab

C = ?

c = ?

Situation I: Angle A is obtuse - EXAMPLE Given a triangle with angle A = 120°, side a = 22

cm and side b = 15 cm, find the other dimensions.

A B

a = 2215 = b

C

c

120°

Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines:

SOLUTION22

sin120

15

sin B15sin120 22sin B

B sin 1 15sin12022

36.2

22

sin120

c

sin 23.8c sin120 22sin 23.8

c 22sin 23.8sin120

10.3cm

Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3 cm

Situation II: Angle A is acute

If angle A is acute there are SEVERAL possibilities.

Side ‘a’ may or may not be long enough to reach side ‘c’. We calculate the height of the altitude from angle C to side c to compare it with side a.

A B ?

b

C = ?

c = ?

a

Situation II: Angle A is acute First, use SOH-CAH-TOA to find h:

A B ?

b

C = ?

c = ?

a

h

sin A h

bh bsin A

Then, compare ‘h’ to sides a and b . . .

DIFFERENT TRIANGLES

If a < h, then NO triangle exists with these dimensions.

A B ?

b

C = ?

c = ?

a

h

If h < a < b, then TWO triangles exist with these dimensions.

A B

b

C

c

ah

If we open side ‘a’ to the outside of h, angle B is acute

AB

b

C

c

a h

If we open side ‘a’ to the inside of h, angle B is obtuse.

If h < b < a, then ONE triangle exists with these dimensions.

A B

b

C

c

a

h

Since side a is greater than side b, side a cannot open to the inside of h, it can only open to the outside, so there is only 1 triangle possible!

If h = a, then ONE triangle exists with these dimensions.

AB

b

C

c

a = h

If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions.

EXAMPLE

Given a triangle with angle A = 40°, side a = 12 cm and side b = 15 cm, find the other dimensions.

A B ?

15 = b

C = ?

c = ?

a = 12

h

40°

SOLUTION

Angle B = 53.5° Angle C = 86.5° Side c = 18.6

Angle B = 126.5° Angle C = 13.5° Side c = 4.4

THE AMBIGUOUS CASE - SUMMARY

if angle A is acute

find the height,h = b*sinA

if angle A is obtuse if a < b no solution

if a > b one solution

if a < h no solution

if h < a < b 2 solutions one with angle B acute, one with angle B obtuse

if a > b > h 1 solution

If a = h 1 solution angle B is right

(Ex I)

(Ex II-1)

(Ex II-2)

LAW OF SINES

Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.

AAS ASA SSA (the

ambiguous case)

a

sin A

b

sin B

c

sin C

STUDENT GUIDED PRACTICE

Do problems 7-12 in your book page 410

HOMEWORK

Do problems 13-17, 27-29 in your book page 410

CLOSURE

Today we learned about the law of sines

Next class we are going to learn about law of cosines