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Math III Accelerated

Chapter 13Trigonometric Ratios

and Functions

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Warm Up 13.5

1. Solve triangle ABC.

2. Solve triangle ABC with B = 41°, C = 90°, and c = 22.

b = 13A

B

C

ac

33°

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13.5 Apply the Law of Sines

Objective: Solve triangles that have no right angle.

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The Law of Sines

Law of Sines

In any triangle ABC, with sides a, b, and c,

.sinsinsin Cc

Bb

Aa

.sinsinsincC

bB

aA

C

BA

ab

c

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Use of Law of Sines

Use Law of Sines when we know:

One side and two angles SAA or ASA

Two sides and an angle opposite SSA This may lead to more than one solution.

Recall from geometry – SAA and ASA prove triangle congruence, but SSA doesn’t.

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Example 1 Solve triangle ABC with A = 28°, B = 102°,

and a = 8.

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The Ambiguous Case - SSA

In the ambiguous (SSA) case, there are three possibilities:

1. There is no triangle with the given information.

2. There is exactly one triangle with the given information.

3. There are two triangles, one acute and one obtuse, with the given information.

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The Ambiguous Case - SSA

Given two sides and an angle opposite one of them, several possibilities exist:

No solution - side too shortto make a triangle

One solution - side equals altitude 20°

10 3.42

20°

10 1

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The Ambiguous Case - SSA

Two possible triangles could result.

One unique solution - the opposite sideis longer thanadjacent side.

Solving for A could give either an acute

or obtuse angle!

Solving for A could give either an acute

or obtuse angle! 20°

105 5

AA'

20°

10 13.42

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Example 2 Solve triangle ABC with A = 94°, a = 18,

and c = 13.

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Example 3 Solve triangle ABC with A = 77°, a =

6.1, and b = 9.

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Checkpoints 1, 2, & 3 Solve triangle ABC.

1. C = 14°, B = 117°, b = 21

2. A = 56°, a = 24, b = 16

3. B = 122°, b = 5, a = 8

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Example 4 Solve triangle ABC with A = 30°, a = 10,

and b = 15.

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Area of a Triangle

In any triangle ABC, with sides a, b, and c, the area is given by:

A = ½ ab sin C

A = ½ ac sin B

A = ½ bc sin A

C

BA

ab

c

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Example 5 A piece of land is bordered by three roads as

shown. Find the area of the land.B

A

C

1.4 mi

2.3 mi

78.1°

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Checkpoints 4 & 5

4. Solve triangle ABC with A = 35°, a = 11, and b = 14.

5. Suppose the side lengths in Example 5 are 4.6 miles and 2.8 miles. Find the area.

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Homework 13.5

Practice 13.5