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)

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?

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

Area of Triangle - Angle A C (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

Area of Triangle - Angle B y (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

Area of Triangle - Angle C B 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

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

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

Regents Prep In ΔABC, mA = 120, b = 10, and c = 18. What is the area of ΔABC to the nearest square inch? 53 2. 78 3. 90 4. 156

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

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

The Product Rule

The Product Rule

Dilating the Unit Circle y 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