Addition and Subtraction Formulas for Sine and Cosine

Verbalizing/Recalling the Sum Formulas

The sum formula for the cosine gives the cosine of a sum in terms of the sine and cosine of the addends:

Here's a way to recall, from memory, the formula for $\,\cos(a + b)\,$:

• first write the pattern:   $\cos\,\cos\ \ \ \ \sin\,\sin$
• recall: ‘sine is the same’; cosine is different:
  thus the plus sign in $\,\cos(a \color{red}{+} b)\,$ gets changed to a minus sign:
  $\cos\,\cos\ \color{red}{-}\ \sin\,\sin$
• put the addends in place (twice), in the same order as they appear in $\,\cos(a+b)\,$:
  $\cos a\,\cos b\ \color{red}{-}\ \sin a\,\sin b$

The sum formula for the sine gives the sine of a sum in terms of the sine and cosine of the addends:

Here's a way to recall, from memory, the formula for $\,\sin(a + b)\,$:

• first write the pattern:   $\sin\,\cos\ \ \ \ \cos\,\sin$
• recall: ‘sine is the same’:
  thus the plus sign in $\,\sin(a \color{red}{+} b)\,$ remains a plus sign:
  $\sin\,\cos\ \color{red}{+}\ \cos\,\sin$
• put the addends in place (twice), in the same order as they appear in $\,\sin(a+b)\,$:
  $\sin a\,\cos b\ \color{red}{+}\ \cos a\,\sin b$

There are similar verbalizations and memory recall methods for the difference formulas.

Example: Using the Sum Formulas

Let's use some special angles for an example.
You know that $\cos 90^\circ = 0\,$ and $\sin 90^\circ = 1\,.$
Do the sum formulas give these results? $$ \begin{align} \cssId{s97}{\cos 90^\circ = \cos (30^\circ + 60^\circ)} &\cssId{s98}{\overset{\text{?}}{=}} \cssId{s99}{\cos 30^\circ\cos 60^\circ - \sin 30^\circ\sin 60^\circ}\cr &\cssId{s100}{= \ \ \ \frac {\sqrt 3}2\ \ \ \cdot\ \ \frac{1}{2}\ \ - \ \ \frac{1}{2}\ \cdot\ \ \frac {\sqrt 3}2\ \ =\ \ 0\ \ \ \ \text{Yep!}}\cr\cr\cr \cssId{s101}{\sin 90^\circ = \sin (30^\circ + 60^\circ)} &\cssId{s102}{\overset{\text{?}}{=}} \cssId{s103}{\sin 30^\circ\cos 60^\circ + \cos 30^\circ\sin 60^\circ}\cr &\cssId{s104}{= \ \ \ \frac{1}{2}\ \ \ \cdot\ \ \frac{1}{2}\ \ + \ \ \frac{\sqrt 3}{2}\ \cdot\ \ \frac {\sqrt 3}2\ \ =\ \ \frac 14 + \frac 34\ \ =\ \ 1\ \ \ \ \text{Yep!}}\cr\cr\cr \end{align} $$

You should do similar examples (say, writing $\,30^\circ = 90^\circ - 60^\circ\,$) to give some confidence in the difference formulas.

Proving the Sum Formulas for Sine and Cosine

An identity is a mathematical sentence that is always true.
The sum formulas given above can't be proved using the simple strategies outlined in Verifying Trigonometric Identities.
They require some cleverness!

When I was talking about these identities one day, my genius husband (Ray) drew a sketch which gives both formulas.
The sketch is shown below, together with step-by-step details of how to get the sum formulas from the sketch. I love it!
Put the origin at point $\,A\,$; assume both $\,a\,$ and $\,b\,$ are measured in degrees.

Start with right triangle $\,\triangle ABC\,$ (mostly yellow). It has acute angle $\,a\,$ and hypotenuse of length $\,1\,.$ Thus: bottom leg: $\,\cos a\,$ side leg: $\,\sin a\,$ Stack a right triangle $\,\triangle ADE\,$ (mostly green) on the hypotenuse of the yellow triangle. It has acute angle $\,b\,$ and hypotenuse of length $\,1\,.$ Thus: bottom leg: $\,\cos b\,$ side leg: $\,\sin b\,$ the blue right triangle: since $\,\overline{DF}\, ||\, \overline{AB}\,,$ $\,\angle CDF = a\,$ hypotenuse: $\,1 - \cos b\,$ Thus: bottom leg: $\,(1-\cos b)(\cos a)\,$ side leg: $\,(1-\cos b)(\sin a)\,$ the purple right triangle: by vertical angles, $\,\angle ADG = a\,$ thus: $\,\angle EDG = 90^\circ - a\,$ thus: $\,\angle DEG = a\,$ hypotenuse: $\,\sin b\,$ thus, bottom leg: $\,\sin b\,\sin a\,$ thus, side leg: $\,\sin b\,\cos a\,$

With all side lengths in place, the sum formulas are now easy: $$ \begin{alignat}{2} \cssId{s146}{\cos(a+b)} &\cssId{s147}{\ =\ \cos a - (1-\cos b)(\cos a) - \sin b\,\sin a} &\qquad&\cssId{s148}{\text{($x$-value of point $\,E\,$) }}\cr\cr &\cssId{s149}{\ =\ \cos a - \cos a + \cos b\,\cos a - \sin b\,\sin a}&\qquad&\cssId{s150}{\text{(distributive law)}}\cr\cr &\cssId{s151}{\ =\ \cos a\,\cos b - \sin a\,\sin b}&\qquad&\cssId{s152}{\text{(cancel; commutative property of multiplication)}}\cr\cr\cr\cr \cssId{s153}{\sin(a+b)} &\ \cssId{s154}{=\ \sin a - (1-\cos b)(\sin a) + \sin b\,\cos a} &\qquad&\cssId{s155}{\text{($y$-value of point $\,E\,$) }}\cr\cr &\cssId{s156}{\ =\ \sin a - \sin a + \cos b\,\sin a + \sin b\,\cos a}&\qquad&\cssId{s157}{\text{(distributive law)}}\cr\cr &\cssId{s158}{\ =\ \sin a\,\cos b + \cos a\,\sin b}&\qquad&\cssId{s159}{\text{(cancel; commutative property of multiplication)}} \end{alignat} $$

For the sketch given here, all angles are acute: $$ \cssId{s161}{0 < a < 90^\circ}\,,\qquad \cssId{s162}{0 < b < 90^\circ}\,,\qquad \cssId{s163}{\,0 < a+b < 90^\circ}\,$$ This proof can be extended for other angles.
Or, a proof for all real numbers can be found in standard texts.

Proving the Difference Formulas for Sine and Cosine

Since subtraction is a special kind of addition, the difference formulas follow easily from the sum formulas. $$ \begin{alignat}{2} \cssId{s168}{\cos(a-b)}\ \ &\cssId{s169}{= \ \ \cos (a + (-b))} &\qquad&\cssId{s170}{\text{(to subtract $b$, add the opposite)}}\cr &\cssId{s171}{= \ \ \cos(a)\,\cos(-b) - \sin(a)\,\sin(-b)} &&\cssId{s172}{\text{(sum formula for cosine)}}\cr &\cssId{s173}{= \ \ \cos(a)\,\cos(b) - \sin(a)\bigl(-\sin(b)\bigr)}&&\cssId{s174}{\text{(cosine is even; sine is odd)}}\cr &\cssId{s175}{= \ \ \cos a\,\cos b + \sin a\,\sin b} &&\cssId{s176}{\text{(simplify)}}\cr\cr\cr\cr \cssId{s177}{\sin(a-b)}\ \ &\cssId{s178}{= \ \ \sin (a + (-b))} &\qquad&\cssId{s179}{\text{(to subtract $b$, add the opposite)}}\cr &\cssId{s180}{= \ \ \sin(a)\,\cos(-b) + \cos(a)\,\sin(-b)} &&\cssId{s181}{\text{(sum formula for sine)}}\cr &\cssId{s182}{= \ \ \sin(a)\,\cos(b) + \cos(a)\bigl(-\sin(b)\bigr)}&&\cssId{s183}{\text{(cosine is even; sine is odd)}}\cr &\cssId{s184}{= \ \ \sin a\,\cos b - \cos a\,\sin b} &&\cssId{s185}{\text{(simplify)}} \end{alignat} $$