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# Sum and difference formulas

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what is tan7π/12 using the tam sum or difference formulas?

What is sin105 using the sin sum or difference formulas?

May 15, 2018

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$$\tan\big(\frac{7\pi}{12}\big)=\tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)$$

And the sum of two angles formula for tan is:

$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$         so......

$$\tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=\frac{\tan\frac{\pi}{3}+\tan\frac{\pi}{4}}{1-\tan\frac{\pi}{3}\tan\frac{\pi}{4}}$$

And we know   $$\tan\frac\pi3=\sqrt3$$   and   $$\tan\frac\pi4=1$$    so...

$$\tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=\frac{ \sqrt3 +1}{1-(\sqrt3)(1)}\\~\\ \tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=\frac{ \sqrt3 +1}{1-\sqrt3}$$

Multiply numerator and denominator by  $$(1+\sqrt3)$$ .

$$\tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=\frac{ (\sqrt3 +1)(1+\sqrt3)}{(1-\sqrt3)(1+\sqrt3)}\\~\\ \tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=\frac{ 2\sqrt3+4}{-2}\\~\\ \tan\big( \frac{\pi}{3}+\frac{\pi}{4} \big)=-\sqrt3-2\\~\\ \tan\frac{7\pi}{12}=-\sqrt3-2\\~\\ \text{________________________}$$

sin( 105° )  =  sin( 45° + 60° )

And the sum of two angles formula for sin is:

sin(α + β)  =  sin α cos β + cos α sin β         so....

sin(45° + 60°)  =  sin 45° cos 60° + cos 45° sin 60°

sin(45° + 60°)  =  $$\big(\frac{\sqrt2}{2}\big)\big(\frac{\sqrt3}{2}\big)+\big(\frac{\sqrt2}{2}\big)\big(\frac12\big)$$

sin(45° + 60°)  =  $$\frac{\sqrt6}{4}+\frac{\sqrt2}{4}$$

sin(45° + 60°)  =  $$\frac{\sqrt6+\sqrt2}{4}$$

sin 105°   =   $$\frac{\sqrt6+\sqrt2}{4}$$

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May 15, 2018