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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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
The Product Rule
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
The Product Rule
Aim: Finding Area of Triangle Course: Alg. 2 & Trig.
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).