1. What is the area of the triangle? A = 1/2 bh b = cos = x h = sin = y
Aim: How do we find the area of a triangle when given two adjacent sides and the included angle? -1 1 y
Do Now:
(cos, sin) 
cos x
What is the area of the triangle?
A = 1/2 bh
b = cos = x
h = sin = y
A = 1/2 (cos)(sin)
 = 60º
A = 1/2 (cos60)(sin60)

2. Un-unit circle
 is any angle in standard position with (x, y) any point on the terminal side of  and
r  1 y x 1 -1
unit circle
How long is r?

3. Model Problem
(-3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of . 3 4 r
= 5
Q II

4. Area of Triangle - Angle AC
(b cos A, b sin A)
(x, y) y h a b A c A B x
base
Area = 1/2 base · h
h = ?
base · sin A
If you know the value of c and b
and the measure of A, then
Area of ∆ABC = 1/2 c • b sinA

5. Area of Triangle - Angle By
(c cos B, c sin B) A b h c B a B C x
h = ?
c sin B
If you know the value of c and a
and the measure of B, then
Area of ∆ABC = 1/2 a • c sinB

6. Area of Triangle - Angle CB y
(a cos C, a sin C) a c h C C b A x
h = ?
a sin C
If you know the value of a and b
and the measure of C, then
Area of ∆ABC = 1/2 a • b sinC

7. The area of a triangle is equal to one-half
Area of Triangle
The area of a triangle is equal to one-half
the product of the measures of two sides
and the sine of the angle between them.
ex. - acute angle
Find the area of ∆ABC if c = 8, a = 6, mB = 30
ex. - obtuse angle
Find the area of ∆BAD if BA = 8, AD = 6, mA = 150

8. Find the exact value of the area of an equilateral
Model Problem
Find the exact value of the area of an equilateral
triangle if the measure of one side is 4.
each side = 4
each angle = 60º A B C c a b 60

9. Regents Prep
In ΔABC, mA = 120, b = 10, and c = 18. What is the area of ΔABC to the nearest square inch?

10. Find to the nearest hundred the number of
Model Problem
Find to the nearest hundred the number of
square feet in the area of a triangular lot at
the intersection of two streets if the angle of
intersection is 76º10’ and
the frontage along
the streets are 220 feet
and 156 feet. C
156’
76º10’
220’ A B
A = 16,700 square feet

11. The area of a parallelogram is 20. Find the
Model Problem
The area of a parallelogram is 20. Find the
measures of the angles of the parallelogram
if the measures of the two adjacent sides are
8 and 5. A B C D
A=10
Diagonal cuts parallelogram
into 2 congruent triangles,
each with area of 10. 8 5 x
180 – x
sinA = 1/2
mA = 30º
mC = 30º
mB & D = (x – 30º)=150º

12. The Product Rule

13. The Product Rule

14. Dilating the Unit Circley 3 2
(3cos, 3sin) 3 -1
(2cos, 2sin) 2 1  -3 -2 -1 2 3 x -1
Prove that the length of
the hypotenuse is equal
to the coefficient common to the coordinate points (x,y). -2 -3