Selasa, 01 April 2008

Compound Angles, Multiple and Submultiple angles

COMPOUND ANGLES

\sin(A \pm B)=\sin A \cos B \pm \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 \pm B)=\frac{\tan A \pm \tan B}{1 \pm \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}\,



Prove that \sin(A+B) \sin(A-B)=\sin^2 A-\sin^2 B\, and \cos (A+B) \cos(A-B)=\cos^2 A-\sin^2 B,\,

Find the value of \tan [\frac{\pi}{4}+A]\,

Find the value of \cos 105^\circ,\sin 75^\circ\,

what is the value of \frac{\tan 40+\tan 20}{\cot 45-\cot 50 \cot 70}\,

Show that \cos 40+\cos 80+\cos 160 =0\,

Prove that \tan 50 =\tan 40 +2\tan 10 \,

Prove that \cos^2 A+\cos^2 B-2\cos A \cos B \cos (A+B)=\sin^2 (A+B)\,

Prove that \sin^2 \theta+\sin^2 (\theta+60)+\sin^2 (\theta-60)=\frac{3}{2}\,

IF \frac{m+1}{m-1}=\frac{\cos (\alpha-\beta)}{\sin (\alpha+\beta)}\, then prove that m=\tan [\frac{\pi}{4}+\alpha] \tan [\frac{\pi}{4}+\beta]\,

In a triangle ABC if \cot A+\cot B+\cot C=\sqrt{3}\, then show that the triangle is equilateral.

A+B+C=180^\circ\,, prove that \tan A+\tan B+\tan C=\tan A \tan B \tan C\,

If \tan \beta=\frac{n\tan\alpha}{1+(1-n)\tan^2 \alpha}\, then show that \tan (\alpha-\beta)=(1-n)\tan \alpha\,

If A+B=45, prove that (1+\tan A)(1+\tan B)=2\,.Hence show that \tan \frac{45}{2} =\sqrt{2}-1\,

Prove that \tan (A-B)+\tan (B-C)+\tan (C-A)=\tan (A-B) \tan (B-C) \tan (C-A)\,

Prove that\tan (\theta-\frac{3\pi}{4}) \tan (\frac{7\pi}{4}+\theta)+1=0\,

Show that \cos^2 \theta+\cos^2 (60+\theta)+\cos^2 (60-\theta)=\frac{3}{2}\,

Show that \cos A+\cos (240-A)+\cos (240+A)=0\,

Multiple and Submultiple angles

1. \sin 2A=2\sin A \cos A,\sin A=2\sin \frac{A}{2}\cos \frac{A}{2}\,

2. \sin 2A=\frac{2\tan A}{1+\tan^2 A},\sin A=\frac{2\tan \frac{A}{2}}{1+\tan^2 \frac{A}{2}}\,

3. \sin 3A=3\sin A-4\sin^3 A\,

4. \cos 2A=\cos^2 A-\sin^2 A=2\cos^2 A-1=1-2\sin^2 A\,

5. \cos A=\cos^2 \frac{A}{2}-\sin^2 \frac{A}{2}=2\cos^2 \frac{A}{2}-1=1-2\sin^2 \frac{A}{2}\,

6. \cos 2A=\frac{1-\tan^2 A}{1+\tan^2 A},\cos A=\frac{1-\tan^2 \frac{A}{2}}{1+\tan^2 \frac{A}{2}}\,



7. \cos 3A=4\cos^3 A-3\cos A\,

8. \tan 2A=\frac{2\tan^2 A}{1-\tan^2 A},\tan A=\frac{2\tan \frac{A}{2}}{1-\tan^2 \frac{A}{2}}\,

9. \tan 3A=\frac{3\tan A-\tan^3 A}{1-3\tan^2 A}\,

Prove that \frac{\cos 3A+\sin 3A}{\cos A-\sin A}=1+2\sin 2A\,

Show that \cos^6 A-\sin^6 A=\cos 2A[1-\frac{\sin^2 2A}{4}]\,

Prove that \cot (\frac{\pi}{4}-\theta)=\frac{\cos 2\theta}{1-\sin 2\theta}\,. Hence find the value of \cot 15^\circ\,

If \tan A=\frac{1-\cos B}{\sin B}\,,then prove that \tan 2A=\tan B\,

Prove that \cos (\frac{\pi}{11}) \cos (\frac{2\pi}{11}) \cos (\frac{3\pi}{11}) \cos (\frac{4\pi}{11}) \cos (\frac{5\pi}{11})=\frac{1}{32}\,

Prove that [1+\cos \frac{\pi}{8}][1+\cos \frac{3\pi}{8}][1+\cos \frac {5\pi}{8}][1+\cos \frac{7\pi}{8}]=\frac{1}{8}\,

Prove that \sin A \sin [\frac{\pi}{3}+A] \sin [\frac{\pi}{3}-A]=\frac{1}{4} \sin 3A\,.Hence show that \sin \frac{\pi}{9} \sin \frac{2\pi}{9} \sin \frac{3\pi}{9} \sin \frac{4\pi}{9}=\frac{3}{16}\,

Prove that 16\cos^5 \theta-20\cos^3 \theta+5\cos \theta=\cos 5\theta\,

If m\tan (\theta-30)=n\tan (\theta+120)\, show that \cos 2\theta=\frac{m+n}{2(m-n)}\,

Prove that \sin^4 \frac{\pi}{8}+\sin^4 \frac{3\pi}{8}+\sin^4 \frac{5\pi}{8}+\sin^4 \frac{7\pi}{8}=\frac{3}{2}\,

Prove that \cot \theta+\cot (60+\theta)-\cot (60-\theta)=3\cot 3\theta\,

Transformations

For all C,D \in R\,

1. \sin C+\sin D=2\sin \frac{C+D}{2} \cos \frac{C-D}{2}\,

2. \sin C-\sin D=2\cos \frac{C+D}{2} \sin \frac{C-D}{2}\,

3. \cos C+\cos D=2\cos \frac{C+D}{2} \cos \frac{C-D}{2}\,

4. \cos C-\cos D=-2\sin \frac{C+D}{2} \sin \frac{C-D}{2}\,

5. 2\sin A \cos B=\sin (A+B)+\sin (A-B)\,

6. 2\cos A \sin B=\sin (A+B)-\sin (A-B)\,

7. 2\cos A \cos B=\cos (A+B)+\cos (A-B)\,

8.-2\sin A \sin B=\cos (A+B)-\cos(A-B)\,








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