Question

Given sinx + cosx = $\frac{5}{4}$. If 1 + 2sinxcosx = a, find the value of 32 a.

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Answer 1

We want to find 1+2sinxcosx and sinx + cosx is given. By doing some operations we should get 1+2sinxcosx. We get 2sinxcosx as one term in the expansion of $(sinx+cosx{)}^2$. This is one of the motivations to consider $(sinx+cosx{)}^2$. If not this, by seeing, 1 and 2 sinx cosx, we should be able to guess 1 can be expressed as

$si{n}^{2}x+co{s}^{2}x$

Either way, we have to consider

$(sinx+cosx{)}^2$

$(sinx+cosx{)}^2$ = ${\left(\frac{5}{4}\right)}^2$ = $\frac{25}{16}$

$\Rightarrow$ $si{n}^{2}x+co{s}^{2}x$ + 2sinx cosx = $\frac{25}{16}$ $si{n}^{2}x+co{s}^{2}x$ = 1

$\Rightarrow$ 1 + 2sinx cosx = $\frac{25}{16}$ = a

$\Rightarrow$ 32a = 32 $\times$ $\frac{25}{16}$ = 50

Key steps/ concepts: (1) Guessing the operation to be performed to get

1 + 2sinxcosx from sinx + cosx (2) $si{n}^{2}x+co{s}^{2}x$ = 1