## 三角函數 `$ \sin(-a) = -\sin a $` `$ \cos(-a) = \cos a $` `$ \sin(\frac{\pi}{2}-a) = cosa $` `$ \cos(\frac{\pi}{2}-a) = \sin a $` `$ sin(\frac{\pi}{2}+a) = \cos a $` `$ cos(\frac{\pi}{2}+a) = -\sin a $` `$ \sin(\pi -a) =\sin a $` `$ \cos(\pi -a) =-\cos a $` `$ \sin(\pi +a) = -\sin a $` `$ \cos(\pi +a) = -\cos a $` ## 和差公式 1. `$ \sin (A+B) = \sin A \cos B +\cos A\sin B $` 2. `$ \cos (A+B) = \cos A \cos B -\sin A\sin B $` 3. `$ \sin (A-B) = \sin A \cos B -\cos A\sin B $` 4. `$ \cos (A-B) = \cos A \cos B +\sin A\sin B $` 5. `$ tan(A+B) = \frac {tanA+tanB} {1-tanAtanB} $` 6. `$ cot(A+B) = \frac {cotAcotB-1}{cotA+cotB} $` 7. `$ tan(A-B) = \frac {tanA-tanB} {1+tanAtanB} $` 8. `$ cot(A+B) = \frac {cotAcotB+1}{cotB-cotA} $` ## 倍角公式 `$ tan2A = \frac{2tanA}{1-tan^2A} $` `$ sin2A = 2sinAcosA $` `$ cos2A = cos^2A-sin^2 = 2cos^2A-1 =1-2sin^2A $` ## 三倍角公式 `$ sin3A = 3sinA-4(sinA)^3 $` `$ cos3A = 4(cosA)^3-3cosA $` `$ tan3A= tanA\cdot tan(\frac{\pi}{3}+a)\cdot tan(\frac{\pi}{3}-a) $` ## 半角公式 `$ sin(\frac A 2) = \sqrt{\frac {1-cosA}{2}} $` `$ cos(\frac A 2) = \sqrt{\frac {1+cosA}{2}} $` `$ tan(\frac A 2) =\sqrt{\frac {1-cosA}{1+cosA}} $` `$ cot(\frac A 2) =\sqrt{\frac {1+cosA}{1-cosA}} $` ## 和差化積 `$ sin a + sin b = 2sin \frac{a+b}{2} cos \frac {a-b}{2} $` `$ sin a -sinb = 2cos \frac{a+b}{2} sin \frac {a-b}{2} $` `$ cos a+cos b = 2cos \frac{a+b}{2} cos \frac {a-b}{2} $` `$ cos a-cos b = -2sin \frac{a+b}{2} sin \frac {a-b}{2} $` `$ tan a+ tan b=\frac {sin(a+b)}{cos a cos b} $` ## 積化和差 `$ sin a sin b = - \frac {1} {2} [cos(a+b)-cos(a-b)] $` `$ cos a cos b = \frac {1} {2} [cos(a+b)+cos(a-b)] $` `$ sin a cos b = \frac{1} {2} [sin(a+b)+sin(a-b)] $` `$ cos a sin b = \frac{1} {2} [sin(a+b)-sin(a-b)] $` ## 萬能公式 `$ sin a = \frac {2 tan \frac {a} {2}} {1+(tan \frac a 2)^2} $` `$ cos a = \frac {1-(tan \frac a 2)^2} {1+(tan \frac a 2)^2} $` `$ tan a = \frac {2 tan \frac {a} {2}} {1-(tan \frac a 2)^2} $` ## 其他 `$ csc a = \frac 1 {sin a} $` `$ sec a =\frac 1 {cos a} $` `$ csc^2a = cot^2a+1 $` `$ sec^2a = tan^2a+1 $` `$ $` ***** *****