ত্রিকোণমিতিক অনুপাত

অধ্যায়: ৯


1

  • $\sin \longleftrightarrow \operatorname{cosec}$
  • $\operatorname{cos} \longleftrightarrow \sec$
  • $\tan \longleftrightarrow \cot$

2

  • $\sin ^{2} \theta+\cos ^{2} \theta=1$
  • $\sec ^{2} \theta-\tan ^{2} \theta=1$
  • $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$
  • $\tan \theta=\frac{\sin \theta}{\cos \theta}$
  • $\cot \theta=\frac{\cos \theta}{\sin \theta}$
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \theta & 0^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} & 180^{\circ} & 270^{\circ} & 360^{\circ} \\ \hline \sin \theta & 0 & \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{\sqrt{3}}{2} & 1 & 0 & -1 & 0 \\ \hline \cos \theta & 1 & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} & 0 & -1 & 0 & 1 \\ \hline \tan \theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & 0 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & 0 \\ \hline \operatorname{cosec} \theta & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & 2 & \sqrt{2} & \frac{2}{\sqrt{3}} & 1 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & -1 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} \\ \hline \sec \theta & 1 & \frac{2}{\sqrt{3}} & \sqrt{2} & 2 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & -1 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & 1 \\ \hline \cot \theta & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & \sqrt{3} & 1 & \frac{1}{\sqrt{3}} & 0 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} & 0 & \begin{array}{c} \text { Not } \\ \text { Defined } \end{array} \\ \hline \end{array} $$
  • sin (90°- θ) = cos θ
  • cos (90°- θ) = sin θ
  • cosec (90°- θ) = sec θ
  • sec (90°- θ) = cosec θ
  • tan (90°- θ) = cot θ
  • cot (90°- θ) = tan θ

$\tan \mathrm{A}+\sin \mathrm{A}=\mathrm{m}$ এবং $\tan \mathrm{A}-\sin \mathrm{A}=\mathrm{n}$
এবং $\mathrm{p}=\tan ^{2} \theta-(1+\sqrt{3}) \tan \theta$.

  • প্রমাণ কর যে, $\tan ^{2} \mathrm{~A} \cdot \sin ^{2} \mathrm{~A}=\mathrm{mn}$.
  • প্রমাণ কর যে, $\mathrm{m}^{2}-\mathrm{n}^{2}=4 \sqrt{\mathrm{mn}}$.
  • $\mathrm{p}+\sqrt{3}=0$ সমীকরণটি প্রমাণ কর ।

$\triangle \mathrm{ABC}$ এ $\angle \mathrm{B}=90^{\circ}, \angle \mathrm{A}=\mathrm{x}-\mathrm{y}, \angle \mathrm{C}=\mathrm{x}+\mathrm{y}, \mathrm{AB}=\sqrt{3}$ এবং $\mathrm{BC}=1$

  • $\mathrm{AC}$ এর দৈর্ঘ্য নির্ণয় কর ।
  • উদ্দীপকের আলোকে $\frac{\operatorname{cosec}^{2} \mathrm{~A}-\sec ^{2} \mathrm{~A}}{\cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~A}}$ এর মান নির্ণয় কর ।
  • $x$ ও $y$ এর মান নির্ণয় কর ।