this is my first question. So, recently I came across a question "$\sin(A) + \cos(A) = 1/4$. Find $\sin(A)\cos(A)$ without a calculator." I was interested and I try to solve it. But unfortunately, I cannot solve it. I think it is about double angles formula $\sin(2A) = 2\sin A\cos A$ but I still didn't know how to solve it. Pls help me.
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14Hint : Square it. – TheSilverDoe Apr 01 '22 at 14:34
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Welcome to MSE. Here's how to ask a good question. Follow these guidelines to get help in this forum. For example, use MathJax to typeset math formulas – jjagmath Apr 01 '22 at 14:40
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If you want to find $A$ directly, this is a particular case of a general question here : https://math.stackexchange.com/questions/213545/solving-trigonometric-equations-of-the-form-a-sin-x-b-cos-x-c – Lelouch Apr 01 '22 at 15:45
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Squaring both sides we have \begin{align} &\sin^2 (A) + 2 \sin (A) \cos (A) + \cos^2(A) = \dfrac{1}{16}\\ \implies &1-\cos ^2 (A) + 2\sin (A) \cos(A) + \cos ^2(A) = \dfrac{1}{16}\\ \implies & 2 \sin (A) \cos(A) = - \dfrac{15}{16}\\ \implies & \sin(A) \cos(A) = -\dfrac{15}{32} \end{align} where the only tool we used was that $\sin^2x = 1 - \cos^2 x$.
Prime Mover
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algevristis
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Not that the OP asked for this extra tid-bit, but I see there are four solutions for $A$ on the interval $[0, 2\pi]$. More solutions occur at shifts of integer multiples of $2 \pi$ of course. – Galen Apr 01 '22 at 15:47
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