If $$\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$$ then prove that $$\sin 2A \quad\text{ is a root of }\quad x^2 - 44x - 36 = 0$$
I have no idea how to solve it. Plz help.
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Let $$\sin{A}+\cos{A}=t$$
$$7=\sin A+\cos A+\frac{1}{\sin A\cos A}+\frac{\sin A+\cos A} {\sin A\cos A}$$
$$\sin 2A =2 \sin A \cos A$$
$$t^2=1+\sin 2A$$
evil999man
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Can u plz elaborate? – Apr 30 '14 at 05:34
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@user36790 Done. – evil999man Apr 30 '14 at 05:36
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Will I have to pair tan & cot and sec & cosec ? – Apr 30 '14 at 05:36
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@user36790 Yes. – evil999man Apr 30 '14 at 05:37
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Why have u taken t?Will i have to substitute sinxcosx in terms of t?Do u want to find the value of t? – Apr 30 '14 at 05:42
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@user36790 It's just a substitution. Form an equation in t and finally substitute everything in terms of $\sin 2A$ – evil999man Apr 30 '14 at 05:46
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The value of sin2x is 8 but it doesn't satisfy the equation? – Apr 30 '14 at 06:03
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@user36790 sin is always between -1 and 1. You can eliminate this root by dividing by sin(2x)-8 in the polynomial. – evil999man Apr 30 '14 at 06:21
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let us continue this discussion in chat – Apr 30 '14 at 06:24
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Plz show me the division,sir? – Apr 30 '14 at 16:31
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@user36790 What equation(s) have you got? – evil999man Apr 30 '14 at 16:50
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@user36790 Post them and I will guide you. – evil999man Apr 30 '14 at 16:54
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I think you have the problem wrong, you can show $\sin 2A$ is a root of $x^2-44x+36$ using Awesome's hint.
EDIT:
If $\sin 2A$ is a root of $x^2-44x+36$, then $$\sin 2A = \frac{44 \pm \sqrt{1936 - 144}}{2}=22\pm 8\sqrt{7}$$
Since $|\sin 2A|\leq 1$, we have that $\sin 2A = 22-8\sqrt{7}$.
Using the hint by Awesome,
\begin{align} 7 &=\sin A + \cos A + \frac{1}{\sin A\cos A} + \frac{\sin A + \cos A}{\sin A\cos A} \\ &= t + \frac{1+t}{\frac{1}{2}(t^2-1)} \\ &= t + \frac{2(1+t)}{t^2-1} \\ &= t + \frac{2}{t-1}. \end{align}
This gives the polynomial $t^2-8t+9=0$, which has roots $t=4 \pm \sqrt{7}$. Since $$(\sin A + \cos A)^2 = 1+\sin 2A \leq 1 + 1= 2,$$ we have $$|\sin A + \cos A| \leq \sqrt{2}.$$
This forces $t=4-\sqrt{7}$. You can now compute $\sin 2A = 22-8\sqrt{7}$ as desired.
jbtimmer
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Are u sure bcuz i ve taken it from a trusted mathbook ensured by the govt education board?Obviously u r right as i ve applied the above mentioned method but in vain. – Apr 30 '14 at 16:28
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