a 5.2 m ladder leans against a wall. the bottom of the ladder is 1.9 m from the wall. what angle...

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Warm up A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest degree)? Cos -1 (1.9/5.2) = 69 o

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Page 1: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Warm up   A 5.2 m ladder leans against a wall. The

bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest degree)?

Cos-1(1.9/5.2) = 69o

Page 2: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Objective: To use the Law of Sines in order to solve oblique triangles

The Law of Sines

Page 3: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Consider the first category, an acute triangle (, , are acute).

Page 4: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

hNow, sin( ) , so that h a sin( )

ah

But sin( ) , so that h c sin( )c

By transitivity, a sin( ) c sin( )

sin( ) sin( )Which means

c a

Create an altitude, h.

Page 5: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Theorem Law of Sines

Page 6: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

The Law of Sines is used when we know any two angles and one side or when we know two sides and an angle opposite one of those sides.

Page 7: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Applying the Law of Sines The Law of Sines may be used when the

known parts of the triangle are:◦ 1. one side and two angles (SAA), (ASA)◦ 2. two sides and an angle opposite one of the

sides (SSA)

Page 8: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Example: In triangle ABC, angle A = 106 o, angle B =

31o and side a = 10 cm. Solve the triangle ABC by finding angle C and sides b and c.(round answers to 1 decimal place).

Page 9: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Use the fact that the sum of all three angles of a triangle is equal to 180 o to write an equation in C. A + B + C = 180 o Solve for C. C = 180 o - (A + B) = 43 o

Solution

Page 10: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Use sine law to write an equation in b. a / sin(A) = b / sin(B) Solve for b. b = a sin (B) / sin(A) = (approximately) 5.4 cm

Use the sine law to write an equation in c.a / sin(A) = c / sin(C)Solve for c. c = a sin (C) / sin(A) = (approximately) 7.1 cm

Solution (cont’d)

Page 11: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Area of an Oblique Triangle

The procedure used to prove the Law of Sines leads to a simple formula for the area of an oblique triangle.

Referring to the triangles below, that each triangle has a height of h = b sin A.

A is acute. A is obtuse.

Page 12: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Area of a Triangle - SASSAS – you know two sides: b, c and

the angle between: A

Remember area of a triangle is ½ base ● height

Base = bHeight = c ● sin A Area K= ½ bc(sinA)

A

B

C

c a

b

h

Looking at this from all three sides:K = ½ ab(sin C) = ½ ac(sin B) = ½ bc (sin A)

Page 13: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Example: Find the area ofgiven a = 32 m, b = 9 m, and

ABC36 .m C

132 9 sin36

2Area m m

284.6Area m

Page 14: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

You can also find the area if you know one side and 2 angles based on the Law of Sines.

so, substitute for b in the last

equation, K = ½ bc(sinA) gives you

Area of a Triangle

C

c

BSin

b

sin

C

Bcb

sin

sin

C

BAcK

sin

sinsin

2

1 2

Page 15: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Find the area of triangle JKL if j=45.7, K=111.1o, and L=27.3o.

Area of a Triangle

Page 16: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

http://www.emathematics.net/trigonometria.php?tr=5

Law of Sines practice

Page 17: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Using the triangle above, A = 50o, B = 65o and a = 12.  Solve the triangle.

Warm up

Page 18: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Lesson 5-7 Law of Sines the Ambiguous Case

Objective: To determine whether a triangle has zero, one or two solutions and solve

using the Law of Sines.

Page 19: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

The Ambiguous Case – SSA

In this case, you may have information that results in one triangle, two triangles, or no triangles.

Page 20: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

SSA – No Solution

Two sides and an angle opposite one of the sides.

Page 21: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

By the law of sines,

sin(57 ) sin( )

15 20

Page 22: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Thus,20 sin(57 )

sin( )15

20 (0.8387)sin( )

15sin( ) 1.1183 Impossible!

Therefore, there is no value for that exists! No Solution!

Page 23: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

SSA – Two Solutions

Page 24: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

By the law of sines,

sin(32 ) sin( )

30 42

Page 25: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

So that,

42 sin(32 )sin( )

3042 (0.5299)

sin( )30

sin( ) 0.7419

48 or 132

Page 26: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Case 1 Case 2

48 32 180

100

132 32 180

16

Both triangles are valid! Therefore, we have two solutions.

Page 27: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Case 1 )sin(100 sin(32 )

c 30

30 sin(100 )c

sin(32 )

30 (0.9848)c

0.5299

c 55.7539

Page 28: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Case 2 sin(16 ) sin(32 )

c 30

30 sin(16 )c

sin(32 )

30 (0.2756)c

0.5299

c 15.6029

Page 29: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

For SSA Triangles:

1. If A < 90°a. a < b

1. a < b(sin A) No Solution2. a = b(sin A) 1 Solution3. a > b(sin A) 2 Solution

b. a ≥ b 1 Solution

2. If A ≥ 90°a. a ≤ b No Solution

b. a > b 1 Solution

Page 30: A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest

Practice Solve the triangle: A = 42°, a = 11, and b =

6