Aim: Finding Area of Triangle Course: Alg. 2 & Trig. Aim: How do we find the area of a triangle when...

Aim: Finding Area of Triangle Course: Alg. 2 & Trig.

Aim: How do we find the area of a triangle when given two adjacent sides and the

included angle?Do Now: y

x

(cos, sin)

-1

1-1

1

1

What is the area of the triangle?cos

A = 1/2 bh

A = 1/2 (cos)(sin)b = cos = x h = sin = y

= 60º A = 1/2 (cos60)(sin60)

A 1

2

1

2

3

2

3

8


Un-unit circle

is any angle in standard position with (x, y) any point on the terminal side of and 22 yxr

4

2

-2

-5

r

(x, y)

0,cottan

0,seccos

0,cscsin

yy

x

x

y

xx

r

r

x

yy

r

r

y

y

x1-1

-1

1

1

sin y

cos xunit circle

r 1

sin 0

cos 0

r y y

r x x


Model Problem

(-3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of .

4

2

-5

(-3, 4)

r

22 yxr 22 4)3( r

525 r

= 5

3

4

3

4

3

4tan

5

3

5

3cos

5

4sin

x

yr

xr

y

180 53.130

126.897

Q II

1 4sin 53.130

5


Area of Triangle - Angle A

y

x

(b cos A, b sin A)

ba h

cA B

C

Area of ∆ABC = 1/2 c • b sinA

h = ?base · sin A

If you know the value of c and band the measure of A, then

Area = 1/2 base · h

A

(x, y)

base


Area of Triangle - Angle B

y

x

(c cos B, c sin B)

cb

a

h

A

CB

Area of ∆ABC = 1/2 a • c sinB

h = ?c sin B

If you know the value of c and aand the measure of B, then

B


Area of Triangle - Angle C

y

x

(a cos C, a sin C)

ca

b

h

B

AC

Area of ∆ABC = 1/2 a • b sinC

h = ?a sin C

If you know the value of a and band the measure of C, then

C


Area of Triangle

The area of a triangle is equal to one-halfthe product of the measures of two sidesand the sine of the angle between them.

Area of ABC 1

2ab sinC

1

2ac sinB

1

2bc sin A

ex. - acute angle

Find the area of ∆ABC if c = 8, a = 6, mB = 301

sin2

A ac B

ex. - obtuse angle

Find the area of ∆BAD if BA = 8, AD = 6, mA = 150

A 1

2(BA)( AD) sin A

1

2(8)(6)(.5) 12

1(6)(8)(.5) 12

2


Model Problem

Find the exact value of the area of an equilateraltriangle if the measure of one side is 4.

A 1

2(4)(4) sin60 8(

3

2) 4 3

each side = 4 each angle = 60º

Area of ABC 1

2ab sinC

A

B

C

c a

b

60

6060


Regents Prep

In ΔABC, mA = 120, b = 10, and c = 18. What is the area of ΔABC to the nearest square inch?

1. 53 2. 78 3. 90 4. 156


Model Problem

Find to the nearest hundred the number of square feet in the area of a triangular lot atthe intersection of two streets if the angle ofintersection is 76º10’ and the frontage alongthe streets are 220 feet and 156 feet.

A 1

2(156)(220) sin76º10'

Area of ABC 1

2( AC)( AB) sinA

A B

C

220’

156’76º10’

A 17160(.9709953424) 16662.28008

A = 16,700 square feet


The area of a parallelogram is 20. Find themeasures of the angles of the parallelogramif the measures of the two adjacent sides are8 and 5.

A B

CD

Model Problem

10 1

2(5)(8) sin A

Area of ABD 1

2( AD)( AB) sinA

x8

5180 – x

Diagonal cuts parallelograminto 2 congruent triangles, each with area of 10.

10 20 sinA

sinA = 1/2 mA = 30º

A=10A=10

mC = 30º mB & D = (x – 30º)=150º


The Product Rule


Dilating the Unit Circley

x

(2cos, 2sin)

-2

2-2

2

2

-1

-1

-1(3cos, 3sin)

-3

-3

3

3

1

3

Prove that the length ofthe hypotenuse is equal to the coefficient common to the coordinate points (x,y).

Aim: Finding Area of Triangle Course: Alg. 2 & Trig. Aim: How do we find the area of a triangle when...

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