Trigonometry/Sum and Difference Formulas

Cosine Formulas

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\cos (A-B)=\cos A\cos B+\sin A\sin B$

$\cos 2A={\cos }^{2}A-{\sin }^{2}A=2{\cos }^{2}A-1=1-2{\sin }^{2}A$

$\cos \frac{A}{2}=\pm \sqrt{\frac{1+\cos A}{2}}$

Sine Formulas

$\sin (a+b)=\sin a\cos b+\cos a\sin b$

$\sin (a-b)=\sin a\cos b-\cos a\sin b$

$\sin 2a=2\sin a\cos a$

$\sin \frac{A}{2}=\pm \sqrt{\frac{1-\cos A}{2}}$

Tangent Formulas

$\tan (a+b)=\frac{\tan (a)+\tan (b)}{1-\tan (a)\tan (b)}$

$\tan (a-b)=\frac{\tan (a)-\tan (b)}{1+\tan (a)\tan (b)}$

$\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}=\frac{2\cot A}{{\cot }^{2}A-1}=\frac{2}{\cot A-\tan A}$

$\tan \frac{A}{2}=\pm \sqrt{\frac{1-\cos A}{1+\cos A}}=\frac{\sin A}{1+\cos A}=\frac{1-\cos A}{\sin A}$

Derivations

• cos(a + b) = cos a cos b - sin a sin b
• cos(a - b) = cos a cos b + sin a sin b

Using cos(a + b) and the fact that cosine is even and sine is odd, we have

            cos(a + (-b)) = cos a cos (-b) - sin a sin (-b)
                          = cos a cos b - sin a (-sin b)
                          = cos a cos b + sin a sin b

• sin(a + b) = sin a cos b + cos a sin b

Using cofunctions we know that sin a = cos (90 - a). Use the formula for cos(a - b) and cofunctions we can write

         sin(a + b) = cos(90 - (a + b))
                    = cos((90 - a) - b)
                    = cos(90 -a)cos b + sin(90 - a)sin b
                    = sin a cos b + cos a sin b

• sin(a - b) = sin a cos b - cos a sin b

Having derived sin(a + b) we replace b with "-b" and use the fact that cosine is even and sine is odd.

      sin(a + (-b)) = sin a cos (-b) + cos a sin (-b) 
                    = sin a cos b + cos a (-sin b)
                    = sin a cos b - cos a sin b

make sure when you are applying this formula you admit one cos to 4 -sin Template:Trigonometry:Navigation