Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Some Trigonometric Measures

(A) Formula for Compound Angles measures   (A+B)

(i) \sin (A+B)=\sin A \cdot \cos B+\cos A \cdot \sin B

(ii) \cos (A+B)=\cos A \cdot \cos B-\sin A\cdot \sin B

(iii)  \tan (A+B)=\dfrac{\tan A+\tan B}{1-\tan A \cdot \tan B}

(iv) \cot (A+B)=\dfrac{\cot A \ast \cot B-1}{\cot A+\cot B}

 (\ast \text { is the multiplication of } \cot A \text { and } \cot B)

(B) Formula of Compound Angles measure  (A-B)

(i) \sin (A-B)=\sin A \cdot \cos B- \cos A \cdot \sin B

(ii)  \cos (A-B)= \cos A \cdot \cos B +\sin A \cdot \sin B

(iii)  \tan (A-B)=\dfrac{\tan A-\tan B}{1+\tan A \cdot \tan B}

(iv)  \cot (A-B)=\dfrac{\cot A \cdot \cot B +1}{\cot A -\cot B}

(C) Submultiple formula of Compound Angles:

(i) \sin (A+B)+\sin (A-B)=2 \sin A \cdot \cos B

(ii) \sin (A+B)-\sin (A-B)=2 \cos A \cdot \sin B

(iii)  \cos (A+B)+\cos (A-B)=2 \cos A \cdot \cos B

(iv) \cos (A+B)-\cos (A-B)=-2 \sin A \cdot \sin B

(D) Formula for Three Angles  (A+B+C)

(i) \begin{aligned} \sin (A+B+C)&=\sin A \cdot \cos B \cdot \cos C+\cos A \cdot \sin B \cdot \cos C+\cos A \cdot \cos B \cdot \sin C-\sin A\cdot \sin B \cdot \sin C \\\text( or)&= \cos A \cdot \cos B \cdot \cos C \left( \tan A+\tan B +\tan C -\tan A \cdot \tan B \cdot \tan C\right) \end{aligned}

(ii)  \begin{aligned}\cos (A+B+C)&= \cos A \cdot \cos B \cdot \cos C-\sin A \cdot \sin B \cdot \cos C-\sin A \cdot \cos B \cdot \sin C-\cos A \cdot \sin B \cdot \sin C\\ \text { or }&= \cos A \cdot \cos B \cdot \cos C \left( 1-\tan A \cdot \tan B-\tan B \cdot \tan C-\tan C \cdot \tan A \right) \end{aligned}

(iii) \tan (A+B+C)=\dfrac{\tan A+\tan B+\tan C-\tan A\cdot \tan B\cdot \tan C}{1-\tan A \cdot \tan B-\tan B \cdot \tan C -\tan C \cdot \tan A}

(iv) \cot (A+B+C)=\dfrac{\cot A \cdot \cot B\cdot\cot C-\cot A-\cot B-\cot C}{\cot A \cdot \cot B +\cot B \cdot \cot C +\cot C \cdot \cot A -1}

(E) Double Angle Formula:

 \begin{aligned} (i) \sin 2A&= 2 \sin A \cdot \cos A \quad(i)\\&=\dfrac{2 \tan A}{1+\tan ^2 A} \quad(ii)\\&=\dfrac{2 \cot A}{\cot^2A+1}\quad(iii) \end{aligned}

  \begin{aligned} (ii) \cos 2A&= \cos^2 A-\sin^2A \quad (i)\\&=2 \cos^2 A-1 \quad (ii)\\&=1-2\sin^2A \quad (iii)\\&=\dfrac{1-\tan^2A}{1+\tan^2A}\quad (iv)\\&=\dfrac{\cot^2 A-1}{\cot^2A-1}\quad (v) \end{aligned}

\begin{aligned} (iii)\tan 2A&=\dfrac{2 \tan A}{1-\tan ^2 A}\quad (i)\\&= \dfrac{2 \cot A}{\cot^2A-1}\quad (ii)\\&=\dfrac{2}{\cot A-\tan A}\quad (iii) \end{aligned}

 \begin{aligned} (iv) \cot 2A &=\dfrac{1-\tan ^2A}{2 \tan^2A}\quad (i)\\&=\dfrac{\cot^2A-1}{2 \cot A}\quad (ii) \\&=\dfrac{\cot A-\tan A}{2}\quad (iii)\end{aligned}

\begin{aligned} (v) \sec 2A&=\dfrac{1+\tan^2A}{1-\tan ^2A}\quad (i)\\&=\dfrac{\cot^2A+1}{\cot^2A-1} \quad (ii) \end{aligned}

 \begin{aligned} (vi)\csc 2A&=\dfrac{1+\tan^2A}{2 \tan A}\quad (i) \\&= \dfrac{\cot^2A+1}{2 \cot A} \quad (ii) \end{aligned}

Note:

(i)  1+\sin 2a = (\cos A-\sin A)^2

(ii) 1-\sin 2A =(\cos A-\sin A)^2

(iii)  1+\cos 2A=2\cos^2A

(iv) 1-\cos 2a=2\sin^2A

(F) Multiple Angle Formula (3A)

 \begin{aligned} (i) \sin 3A&= 3\sin A-4 \sin^3A \quad (i)\\&=3 \cos^2A \cdot \sin A-\sin^3A\quad (ii) \end{aligned}

\begin{aligned} (ii)\cos 3A &= 4\cos^3A-3 \cos A \quad (i)\\&= \cos^3A-3\cos A \cdot \sin ^2 A \quad (ii) \end{aligned}

(iii)  \tan 3A =\dfrac{3 \tan A -\tan^3 A}{1-3 \tan^2A}

(iv)  \cot 3A =\dfrac{\cot^3A-3\cot A}{3 \cot^2A-1}

Multiples Angle (4A)

(i) \sin 4A =4 \sin A \cdot \cos A -8 \sin ^3A \cdot \cos A

(ii) \cos 4A =8 \cos^4A-8\cos^2A+1

(iii)  \tan 4A=\dfrac{4\tan A-4\tan^3A}{1-6\tan^2A +\tan ^4A}

\begin{aligned} (iv) \cot 4A&=\dfrac{1-6\tan^2A+\tan^4A}{4\tan A-4\tan^3A}\quad (i)\\&=\dfrac{\cot^4A-6\cot^2A+1}{4 \cot^3A-4\cot A} \quad (ii) \end{aligned}

Multiple Angles (5A)

(i) \sin 5A =5\sin A-20 \sin^3A+16 \sin^5A

(ii) \cos 5A= 16 \cos ^5 A-20 \cos^3A+5 \cos A

(iii)  \tan 5A=\dfrac{\tan^5A-10\tan^3A+5\tan A}{1-10 \tan^2A+5\tan^4A}

 \begin{aligned} (iv) \cot 5A&=\dfrac{1-10\tan ^2A+5\tan^4 A}{\tan^5A-10\tan^3 A+5 \tan A}\quad (i)\\&=\dfrac{\cot^5A-10\cot^3A+5 \cot A}{1-10 \cot^2A+5\cot^4A}\quad (ii) \end{aligned}

(G) Half Angle Formula \dfrac{A}{2}

(i)  \sin \dfrac{A}{2}=\sqrt{\dfrac{1-\cos A}{2}}

(ii)  \cos \dfrac{A}{2}=\sqrt{\dfrac{1+\cos A}{2}}

\begin{aligned} (iii) \tan \dfrac{A}{2}&=\sqrt{\dfrac{1-\cos A}{1+\cos A}} \quad (i)\\&=\dfrac{\sin A}{1+\cos A}\quad (ii)\\&=\dfrac{1-\cos A}{\sin A}\quad (iii)\\&= \csc A-\cot A \quad (iv) \end{aligned}

\begin{aligned} (iv) \cot \dfrac{A}{2}&= \sqrt{\dfrac{1+\cos A}{1-\cos A}}\quad (i)\\ \\&=\dfrac{\sin A}{1-\cos A}\quad (ii)\\ \\&=\dfrac{1+ \cos A}{\sin A} \quad (iii) \\ \\&=\csc A+\cot A \quad (iv) \end{aligned}

(H) Half Angle Tangent Identities:

(i)  \sin A =\dfrac{ 2 \tan \dfrac{A}{2}}{1+ \tan^2 \dfrac{A}{2}}

(ii)  \cos A=\dfrac{1-\tan^2 \dfrac{A}{2}}{1+\tan^2 \dfrac{A}{2}}

(iii)  \tan A =\dfrac{2 \tan \dfrac{A}{2}}{1-\tan^2\dfrac{A}{2}}

(iv) \cot A=\dfrac{1-\tan^2\dfrac{A}{2}}{2 \tan \dfrac{A}{2}}

(v) \sec A=\dfrac{1+\tan^2 \dfrac{A}{2}}{1-\tan^2 \dfrac{A}{2}}

(vi)  \csc A=\dfrac{1+\tan^2 \dfrac{A}{2}}{2 \tan \dfrac{A}{2}}

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