Geometry IB Date: 4/22/2014 Question: How do we measure the immeasurable? SWBAT use the Law of Sines to solve triangles and problems

Agenda• Bell Ringer: Put up Assigned problems• Go over 8.2/8.3 QuizHW Requests – ws Angle of Elevation and Depression 8.5pg 577 #8-11, 17-21, 23, 24, 38HW: WS old textbook on Law of Sines

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Law of Sines In trigonometry, we can use the

Law of Sines to find missing parts of triangles that are not right triangles.

Law of Sines:In ABC,

sin A = sin B = sin C a b c

B

A

C

cb

a

The Law of Sines!

L.T.: Be able to use the Law of Sines to find unknowns in triangles!

Quick Review:

What does Soh-Cah-Toa stand for?

What kind of triangles do we use this for?

What if it’s not a right triangle? GASP!! What do we do then??

right triangles

hyp

oppsin hyp

adjcos

adj

opptan

The Law of Sines:

Note:

capital letters always stand for __________!

lower-case letters always stand for ________!

Use the Law of Sines ONLY when: you DON’T have a right triangle AND

you know an angle and its opposite side

A

B

C

a

b

cc

C

b

B

a

A sinsinsin

sides

angles

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 .*

Example 1 (con’t)

A C

B

70°

80°a = 12c

b

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

Set up the Law of Sines to find side b:

12sin 80 b sin 70

b 12sin80sin 70

12.6cm

=

Example 1 (con’t)

Set up the Law of Sines to find side c:

12sin 30 csin70

c 12sin 30sin70

6.4cm

A C

B

70°

80°a = 12c

b = 12.630°

=

Example 1 (solution)

Angle C = 30°

Side b = 12.6 cm

Side c = 6.4 cm

A C

B

70°

80°a = 12

c =

6.4

b = 12.630°

Note:

We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

Find p. Round to the nearest tenth.

Example 1a:

Law of Sines

Use a calculator.

Divide each side by sin

Cross products

Answer:

Example 1a:

Law of Sines

Cross products

Divide each side by 7.

to the nearest degree in ,

Example 1b:

Solve for L.

Use a calculator.

Answer:

Example 1b:

a. Find c.

b. Find mT to the nearest degree in RST if r = 12, t = 7, and mT = 76.

Answer:

Answer:

Your Turn:

Solving a Triangle

The Law of Sines can be used to “solve a triangle,” which means to find the measures of all of the angles and all of the sides of a triangle.

We know the measures of two angles of the triangle. Use the Angle Sum Theorem to find

. Round angle measures to the nearest degree and side measures to the nearest tenth.

Example 2a:

Angle Sum Theorem

Subtract 120 from each side.

Add.

Since we know and f, use proportions involving

Example 2a:

To find d:

Law of Sines

Cross products

Substitute.

Use a calculator.

Divide each side by sin 8°.

Example 2a:

To find e:

Law of Sines

Cross products

Substitute.

Use a calculator.

Divide each side by sin 8°.

Answer:

Example 2a:

We know the measure of two sides and an angle opposite one of the sides.

Law of Sines

Cross products

Round angle measures to the nearest degree and side measures to the nearest tenth.

Example 2b:

Solve for L.

Angle Sum Theorem

Use a calculator.

Add.

Substitute.

Divide each side by 16.

Subtract 116 from each side.

Example 2b:

Cross products

Use a calculator.

Law of Sines

Divide each side by sin

Answer:

Example 2b:

Answer:

a. Solve Round angle measures to the nearest degree and side

measures to the nearest tenth.

b. Round angle measures to the nearest degree and side measures to the nearest tenth.

Answer:

Your Turn:

A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is

Draw a diagram Draw Then find the

Example 3:

Since you know the measures of two angles of the

triangle, and the length of a side

opposite one of the angles you

can use the Law of Sines to find the length of the shadow.

Example 3:

Cross products

Use a calculator.

Law of Sines

Answer: The length of the shadow is about 75.9 feet.

Divide each side by sin

Example 3:

A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water?

Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

Your Turn: