10.2The Law of Sines

10.2 The Law of Sines

Objectives:

1. Solve oblique triangles by using the Law of Sines.2. Use area formulas to find areas of triangles.

The Law of Sines

Ex. #1 Solve a Triangle with AAS Information

Solve the triangle.

5.22115sin

27sin4527sin45115sin

27sin45115sin

b

bb

6.30115sin

38sin4538sin45115sin

38sin45115sin

c

cc

3827115180C

Ambiguous Case: SSA

For all these triangles, side a is called the swinging side and side b is called the fixed side.

Ambiguous Case: SSA

Ex. #2 Solve a Trianglewith SSA Information

Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B.

First we draw the triangle. Since the exact shape is unknown, we start with a baseline and estimate a 54° angle. Side c must be across from angle C, so we place side b = 8 adjacent to the angle and c = 5 across from it. Since the angle is unknown, we can attach side c to b, we just don’t know the angle in which they meet.

Side c in this scenario is called the swinging side.



The swinging side can form any angle in a circular motion attached at point A.

Depending on the height h, from A to the baseline, the swinging side could either form one triangle, two triangles, or no triangle.



To find the height h, we will use right triangle trigonometry:

5.654sin88

54sin

hh

h

The height is longer than the swinging side, which cannot reach the baseline to form a triangle.


Lighthouse B is 3 miles east of lighthouse A. Boat C leaves lighthouse B and sails in a straight line. At the moment that the boat is 5 miles from lighthouse B, an observer at lighthouse A notes that the angle determined by the boat, lighthouse A (the vertex), and lighthouse B is 65°. Approximately how far is the boat from lighthouse A at that moment?

h

In this situation, the swinging side is now a = 5 as angle A is given. Because the swinging side is longer than the fixed side c = 3, it will automatically be longer than the height h and form exactly one triangle.


Approximately how far is the boat from lighthouse A at that moment?

94.32565sin3sin

565sin3sin

sin565sin33

sin565sin

1

C

C

C

C

C

To find side b, we must first find angle B. In order to find angle B we must first find angle C.


Approximately how far is the boat from lighthouse A at that moment?

06.8294.3265180 B

miles 5.565sin

06.82sin506.82sin565sin

565sin06.82sin

b

b

bb


Solve triangle ABC when a = 4.8, b = 6, & A = 28°.

Here we must again first find the height in order to see if the swinging side a = 4.8 will reach down and touch the baseline to form a triangle.

8.228sin66

28sin

hh

h


Solve triangle ABC when a = 4.8, b = 6, & A = 28°.With a height of 2.8, the swinging side is long enough to touch the baseline, but instead of forming just one triangle, it forms two triangles.

This occurs whenever the swinging side is longer than the height but shorter than the fixed side.

2.8


Solve triangle ABC when a = 4.8, b = 6, & A = 28°.Since two possible triangles are formed, both need solved. The small numbers next to the sides and angles indicate which triangle we are solving.

93.358.428sin6sin

8.428sin6sin

28sin6sin8.46

sin8.428sin

1

11

1

1

1

B

B

B

B

B 07.11693.35281801C

2.928sin

07.116sin8.407.116sin8.428sin

07.116sin8.428sin

1

1

1

1

c

c

cc


Solve triangle ABC when a = 4.8, b = 6, & A = 28°.To solve the second triangle, we must first use a little geometry and our answer for angle B1 to find angle B2.

Since both legs (formed by the swinging side) are the same length, they form an isosceles triangle. This means the two interior angles are congruent as well. Since we found B1 to be 35.93°, to find B2 we subtract it from 180°.

07.14493.351802B


Solve triangle ABC when a = 4.8, b = 6, & A = 28°.To find angle C2 we simplify must subtract angles A and B2 from 180°.

93.707.144281802C

Finally we can solve for side c2.

4.128sin

93.7sin8.493.7sin8.428sin

93.7sin8.428sin

2

2

2

2

c

c

cc

2.907.116

93.35:1Triangle

1

1

1

cCB

4.193.7

07.144:2Triangle

2

2

2

cCB

Ex. #5 Solve a Trianglewith ASA Information

Two surveyors, standing at points A and B, are measuring a building. The surveyor at point A is 20 feet further away from the building than the surveyor at point B. The angle of elevation of the top of the building from point A is 58°, and the angle of elevation of the top of the building from point B is 72°. How tall is the building?

In order to figure out the height of the building, we will first need to find side a.


How tall is the building? To find side a, we will need the other angles in its triangle.

10872180

108°

141085818014°

11.7014sin

58sin2058sin2014sin

58sin2014sin

a

a

aa


How tall is the building? To find the height of the building h, we will use simple right triangle trigonometry.

108°

14°

ft7.6672sin11.7011.70

72sin

72sin

hh

hah

Area of a Triangle Given SAS

Caution! This formula can only be directly used when given SAS information.

Ex. #6 Find Area with SAS Information

Find the area of the triangle shown in the figure below:

2cm8.6

153sin15

153sin6521

sin21

CabA

Area of a Triangle Given SSS

Use this formula when all sides of a triangle are known. The variable s represents the semi-perimeter which is half the perimeter of the triangle.

Heron’s Formula:

Ex. #7 Find Area withSSS Information

Find the area of the triangle whose sides have lengths of 8, 10, and 14.

First find the semi-perimeter:

163221

1410821

21

cbas

Then plug into Heron’s formula and simplify:

units square 2.391536

26816

1416101681616

csbsassA

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10.2The Law of Sines