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ত্রিকোণমিতিক অনুপাতসমূহ
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  1. ক্যালকুলেটর ব্যবহার না করে মান নির্ণয় কর —
    1. (ক) \(\dfrac{\cos\dfrac{\pi}{4}}{\cos\dfrac{\pi}{6} + \sin\dfrac{\pi}{3}}\)
    2. (খ) \(\tan\dfrac{\pi}{4} + \tan\dfrac{\pi}{6} - \tan\dfrac{\pi}{3}\)
  2. \(\cos\theta = -\dfrac{4}{5}\) এবং \(\pi < \theta < \dfrac{3\pi}{2}\) হলে \(\tan\theta\) এবং \(\sin\theta\) এর মান নির্ণয় কর।
  3. \(\sin A = \dfrac{2}{\sqrt{5}}\) এবং \(\dfrac{\pi}{2} < A < \pi\) এর ক্ষেত্রে \(\cos A\) এবং \(\tan A\) এর মান নির্ণয় কর।
  4. দেওয়া আছে, \(\cos A = \dfrac{1}{2}\)।  যদি \(\cos A\) ও \(\sin A\) একমাত্র চিহ্নবিশিষ্ট হয়, তবে \(\sin A\) ও \(\tan A\) এর মান নির্ণয় কর।
  5. দেওয়া আছে, \(\tan A = -\dfrac{5}{12}\)।  যদি \(\tan A\) ও \(\cos A\) বিপরীত চিহ্নবিশিষ্ট হয়, তবে \(\sin A\) ও \(\cos A\) এর মান নির্ণয় কর।
  6. নিম্নলিখিত অভেদনসমূহ প্রমাণ কর —
    1. (ক) \(\tan A + \cot A = \sec A \csc A\)
    2. (খ) \(\sqrt{\dfrac{1 + \cos\theta}{1 - \cos\theta}} = \csc\theta + \cot\theta = \sqrt{\dfrac{\sec\theta + 1}{\sec\theta - 1}}\)
    3. (গ) \(\sqrt{\dfrac{1 - \sin A}{1 + \sin A}} = \sec A - \tan A\)
    4. (ঘ) \(\sec^4\theta - \sec^2\theta = \tan^4\theta + \tan^2\theta\)
    5. (ঙ) \((\sec\theta - \cos\theta)(\csc\theta - \sin\theta)(\tan\theta + \cot\theta) = 1\)
    6. (চ) \(\dfrac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1} = \tan\theta + \sec\theta\)
  7. যদি \(\csc A = \dfrac{a}{b}\) হয়, যেখানে \(a > b > 0\), তবে প্রমাণ কর যে \(\tan A = \pm\dfrac{b}{\sqrt{a^2 - b^2}}\)।
  8. যদি \(\cos\theta - \sin\theta = \sqrt{2}\sin\theta\) হয়, তবে দেখাও যে \(\cos\theta + \sin\theta = \sqrt{2}\cos\theta\)।
  9. \(\tan\theta = \dfrac{x}{y},\; x \ne y\) হলে,  \(\dfrac{x\sin\theta + y\cos\theta}{x\sin\theta - y\cos\theta}\) এর মান নির্ণয় কর।
  10. \(\tan\theta + \sec\theta = x\) হলে, দেখাও যে \(\sin\theta = \dfrac{x^2 - 1}{x^2 + 1}\)।
  11. \(a\cos\theta - b\sin\theta = c\) হলে, প্রমাণ কর যে \(a\sin\theta + b\cos\theta = \pm\sqrt{a^2 + b^2 - c^2}\)।
  12. মান নির্ণয় কর —
    1. (ক)\( \ \ \sin^2\dfrac{\pi}{6} + \cos^2\dfrac{\pi}{4} + \tan^2\dfrac{\pi}{3} + \cot^2\dfrac{\pi}{6} \)
    2. (খ)\( \ \ 3\tan^2\dfrac{\pi}{4} - \sin^2\dfrac{\pi}{3} - \dfrac{1}{2}\cot^2\dfrac{\pi}{6} + \dfrac{1}{3}\sec^2\dfrac{\pi}{4} \)
    3. (গ)  \( \ \ \tan^2\dfrac{\pi}{4} - \sin^2\dfrac{\pi}{3}\tan^2\dfrac{\pi}{6} - \tan^2\dfrac{\pi}{3}\cos^2\dfrac{\pi}{4}\)
    4. (ঘ) \( \ \ \dfrac{\tan\dfrac{\pi}{3} - \tan\dfrac{\pi}{6}}{1 + \tan\dfrac{\pi}{3}\tan\dfrac{\pi}{6}} + \cos\dfrac{\pi}{3}\cos\dfrac{\pi}{6} + \sin\dfrac{\pi}{3}\sin\dfrac{\pi}{6}\)
  13. সরল কর — \[ \dfrac{1 - \sin^2\dfrac{\pi}{6}}{1 + \sin^2\dfrac{\pi}{4}} \times \dfrac{\cos^2\dfrac{\pi}{3} + \cos^2\dfrac{\pi}{6}}{\cosec^2\dfrac{\pi}{2} - \cot^2\dfrac{\pi}{2}} \div \left( \sin\dfrac{\pi}{3}\tan\dfrac{\pi}{6} \right) + \left( \sec^2\dfrac{\pi}{6} - \tan^2\dfrac{\pi}{6} \right) \]
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