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# math-formula-sheet-doubleangles

In this "math-formula-sheet-doubleangles"  , double and triple angles of trigonometric ratios and their derived formula you can see.

I. Double Angles

1) Sin($$2\Theta$$ )= 2 Sin$$\Theta$$Cos$$\Theta$$

2) Cos($$2\Theta$$)= $$Cos^{2}\Theta - Sin^{2}\Theta$$

= $$2Cos^{2}\Theta - 1$$

= $$1-2Sin^{2}\Theta$$

a)$$Cos^{2}( \Theta ) =\frac{1+Cos(2\Theta)}{2}$$

b)$$Cos( \Theta ) =\sqrt{\frac{1+Cos(2\Theta)}{2}}$$

c)$$Sin^{2}( \Theta ) =\frac{1-Cos(2\Theta)}{2}$$

d)$$Sin( \Theta ) =\sqrt{\frac{1-Cos(2\Theta)}{2}}$$

3) Tan$$(2\Theta) =\frac{2Tan\Theta}{1-Tan^{2}\Theta}$$

4) Sin$$(2\theta) = \frac{2Tan\Theta}{1+Tan^{2}\Theta}$$

5) Cos$$(2\Theta)=\frac{1-Tan^{2}\Theta}{1+Tan^{2}\Theta}$$

II. Their derived formula(half angles)

Replace $$2\Theta$$ by $$\Theta$$ and $$\Theta$$ by $$\frac{\Theta}{2}$$ in above formula, we get

1)Sin$$(\Theta)=2Sin\frac{\Theta}{2}Cos\frac{\Theta}{2}$$

2)Cos$$(\Theta) = Cos^{2}(\frac{\Theta}{2}) - Sin^{2}(\frac{\Theta}{2})$$

=$$2Cos^{2}(\frac{\Theta}{2})-1$$

=1-$$2Sin^{2}(\frac{\Theta}{2})$$

3)Tan$$(\Theta)=\frac{2Tan(\frac{\Theta}{2})}{1-Tan^{2}(\frac{\Theta}{2})}$$

4) $$Tan^{2}(\Theta)=\frac{1-Cos(2\Theta)}{1+Cos(2\Theta)}$$

5)$$Tan(\Theta)=\sqrt{\frac{1-Cos(2\Theta)}{1+Cos(2\Theta)}}$$

III Triple angles of Trigonometric ratios Sin, Cos and Tan as follows:

1) Sin(3$$\Theta)= 3Sin(\Theta)- 4 Sin^{3}(\Theta)$$

a) $$Sin^{3}(\Theta) =\frac{3Sin(\Theta)-Sin(3\Theta)}{4}$$

2) Cos(3$$\Theta)= 4 Cos^{3}(\Theta)- 3 Cos(\Theta)$$

a)$$Cos^{3}(\Theta) =\frac{3Cos(\Theta)+Cos(3\Theta)}{4}$$

3) Tan(3$$\Theta)= \frac{3Tan(\Theta)-Tan^{3}(\Theta)}{1-3Tan^{2}(\Theta)}$$

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