Area A=4sinθ + 2sin2θ,where 0<θ< pi/2.
Find the two possible values of θ, when an area A is 5m^2? On the other hand, if area A is the same but different values of θ, find all possible values for A ?
You need to substitute 5 for A in equation `A = 4 sin theta + 2 sin 2theta` such that:
`5 =4sin theta + 2 sin 2theta`
You need to solve for `theta` the equation above, thus you need to express the equation in terms of `theta` using the formula of double angle such that:
`5 = 4sin theta + 2 (2 sin theta*cos theta)`
`5 = 4sin theta + 4sin theta*cos theta`
You need to factor out `theta ` such that:
`5 = 4sin theta(1 + cos theta)`
You need to divide by 4 both sides such that:
`5/4 = sin theta(1 + cos theta)`
`1.25 = sin theta(1 + cos theta)`
You should write `1.25 = 1*1.25` such that:
`1*1.25 = sin theta(1 + cos theta)`
`sin theta = 1 =gt theta = pi/2`
Notice that `theta = pi/2` is not a convenient value since `theta` needs to be smaller than `pi/2` .
`cos theta + 1 = 1.25 =gt cos theta = 1.25 - 1`
`cos theta = 0.25 =gt theta = cos^(-1) 0.25`
`theta ~~ 75.52^o`
Notice that you may only select `sin theta = 1` , since `sin theta = 1.25 ` becomes a contradiction because the values of sine function are not larger than 1.
Hence, evaluating the solutions to equation yields that `theta` has only one convenient value of `75.52^o` .