Cyndee wants to invest $50,000. Her financial planner advises her to invest in three types of accounts: one paying 4%, one paying 5.5%, and one paying 7% simple interest per year. Cyndee wants to put twice as much in the lowest-yielding, least-risky account as in the highest-yielding account. How much should she invest in each account to achieve a total annual return of $2585?

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brennarfa

23.04.2022

Mathematics

$2\sin 2\theta - 3 \cos2\theta - 3\sin\theta + 3 = \sin\theta(4\cos\theta + 6\sin\theta - 3)$ left-hand side $\begin{aligned} \text{lhs} & = 2\sin 2\theta - 3 \cos2\theta - 3\sin\theta + 3 \\ & = 2\boxed{\sin 2\theta} - 3 \boxed{\cos2\theta} - 3\sin\theta + 3 \end{aligned}$ use identities: $\begin{aligned} \sin 2\theta & = 2\sin\theta\cos\theta \\ \cos 2\theta & = 1 - 2\sin^2\theta \end{aligned}$ (using that form of cosine double angle because there's one 1 cosine on the right-hand side. if we used another form, there'd be too many cosines) $\begin{aligned} \text{lhs} & = 2(2\sin\theta\cos\theta)- 3(1 - 2\sin^2\theta) - 3\sin\theta + 3 \\ & = 4\sin\theta\cos\theta - 3 + 6\sin^2\theta - 3\sin\theta + 3 \\ & =4\sin\theta\cos\theta - \boxed{3} + 6\sin^2\theta - 3\sin\theta + \boxed{3} \end{aligned}$ those +3 and -3 cancel each other out (add to become 0) $\begin{aligned} \text{lhs} & =4\sin\theta\cos\theta + 6\sin^2\theta - 3\sin\theta \end{aligned}$ factor out $\sin\theta$ out of everything $\begin{aligned} \text{lhs} & =4\sin\theta\cos\theta + 6\sin^2\theta - 3\sin\theta \\ & = \sin\theta\left(4\cos\theta + 6\sin\theta - 3\right) \\ & = \text{rhs} \end{aligned}$

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