One way to determine the shaded area Atheta in Question 3 as

One way to determine the shaded area A(theta) in Question 3 as a function of theta is to find the area of the striped rectangle and subtract off the area of the two right triangles. Complete the blanks. What is the area of the large striped rectangle? (This should be an expression involving theta.) Area = What is the total area of the two small triangles? (This should be an expression involving theta.) Area = A(theta)=

Solution

a) Large striped rectnagle has sides of

length = l*cos(theta) + l*cos(theta) +l = 2lcostheta +l

width = lsintheta

Area = l*w = 2lcostheta*lsintheta +l^2sintheta = 2l^2sinthetacostheta + l^2sintheta = l^2sin(2theta) +l^2sintheta

b) Area of triangle 1 triangle = (1/2)*(lcostheta)*(lsintheta) = (l^2/2)costhetasintheta

= (l^2/4)sin2theta

Area of triangle 2 triangles = 2*(l^2/4)sin2theta = (l^2/2)sin2theta

c) A (theta) = Area of large rectangle - Area of 2 triangles

= l^2sin(2theta) +l^2sintheta - (l^2/2)sin2theta

= (l^2/)sintheta + l^2sintheta