HSC Higher Mathematics: Trigonometry Formula Sheet
১. যৌগিক কোণের ত্রিকোণমিতিক অনুপাত (Compound Angles)
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot B + \cot A} \]
\[ \cot(A - B) = \frac{\cot A \cot B + 1}{\cot B - \cot A} \]
২. যোগফল ও গুণফলের রূপান্তর (Sum & Product Transformation)
\[ 2\sin A \cos B = \sin(A+B) + \sin(A-B) \]
\[ 2\cos A \sin B = \sin(A+B) - \sin(A-B) \]
\[ 2\cos A \cos B = \cos(A+B) + \cos(A-B) \]
\[ 2\sin A \sin B = \cos(A-B) - \cos(A+B) \]
\[ \sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]
\[ \sin C - \sin D = 2 \cos\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right) \]
\[ \cos C + \cos D = 2 \cos\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]
\[ \cos C - \cos D = 2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{D-C}{2}\right) \]
৩. গুণিতক ও উপ-গুণিতক কোণ (Multiple Angles)
\[ \sin 2A = 2\sin A \cos A = \frac{2\tan A}{1 + \tan^2 A} \]
\[ \cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1 = \frac{1 - \tan^2 A}{1 + \tan^2 A} \]
\[ 1 + \cos 2A = 2\cos^2 A \quad \implies \quad \cos^2 A = \frac{1 + \cos 2A}{2} \]
\[ 1 - \cos 2A = 2\sin^2 A \quad \implies \quad \sin^2 A = \frac{1 - \cos 2A}{2} \]
\[ \tan 2A = \frac{2\tan A}{1 - \tan^2 A} \]
\[ \sin 3A = 3\sin A - 4\sin^3 A \]
\[ \cos 3A = 4\cos^3 A - 3\cos A \]
\[ \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \]
৪. ত্রিভুজের গুণাবলী (Properties of Triangles)
সাইনের সূত্র (Sine Rule): \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
কোসাইনের সূত্র (Cosine Rule):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
প্রক্ষেপণ সূত্র (Projection Rule):
\[ a = b \cos C + c \cos B, \quad b = c \cos A + a \cos C, \quad c = a \cos B + b \cos A \]
ত্রিভুজের ক্ষেত্রফল (\(\Delta\)):
\[ \Delta = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B = \frac{1}{2}ab \sin C = \sqrt{s(s-a)(s-b)(s-c)} \]
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