﻿ Addition and Subtraction Formulas for Sine and Cosine

# Addition and Subtraction Formulas for Sine and Cosine

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
Note:
For simplicity/brevity,
I sometimes read (say) ‘$\,\sin x\,$’ as just ‘sine ex’,
• PRACTICE (online exercises and printable worksheets)

This section explores four trigonometric identities:

• Addition Formula for Cosine:   $\cos(a\color{blue}{\bf +}b) = \cos a\ \cos b \color{red}{\bf -} \sin a\ \sin b$
• Subtraction Formula for Cosine:   $\cos(a\color{red}{\bf -}b) = \cos a\ \cos b \color{blue}{\bf +} \sin a\ \sin b$
• Addition Formula for Sine:   $\sin(a\color{blue}{\bf +}b) = \sin a\ \cos b \color{blue}{\bf +} \cos a\ \sin b$
• Subtraction Formula for Sine:   $\sin(a\color{red}{\bf -}b) = \sin a\ \cos b \color{red}{\bf -} \cos a\ \sin b$

 SUM AND DIFFERENCE FORMULAS FOR SINE AND COSINE Variety of Names: For all real numbers $\,a\,$ and $\,b\,$: Shorthand Notations Addition Formula for Cosine Sum Formula for Cosine Cosine Addition Formula Cosine Sum Formula $$\cos(a\color{blue}{\bf +}b) = \cos a\ \cos b \color{red}{\bf -} \sin a\ \sin b$$ These two formulas are often presented with this shorthand: $$\cos(a\pm b) = \cos a\ \cos b \ \mp\ \sin a\ \sin b$$ Note that: ‘$\,\pm\,$’ is read as ‘plus or minus’ ‘$\,\mp\,$’ is read as ‘minus or plus’ The plus sign (top left) in $\,\pm\,$ goes with the minus sign (top right) in $\,\mp\,.$ The minus sign (bottom left) in $\,\pm\,$ goes with the plus sign (bottom right) in $\,\mp\,.$ Subtraction Formula for Cosine Difference Formula for Cosine Cosine Subtraction Formula Cosine Difference Formula $$\cos(a\color{red}{\bf -}b) = \cos a\ \cos b \color{blue}{\bf +} \sin a\ \sin b$$ Addition Formula for Sine Sum Formula for Sine Sine Addition Formula Sine Sum Formula $$\sin(a\color{blue}{\bf +}b) = \sin a\ \cos b \color{blue}{\bf +} \cos a\ \sin b$$ These two formulas are often presented with this shorthand: $$\sin(a\pm b) = \sin a\ \cos b \ \pm\ \cos a\ \sin b$$ Note that: ‘$\,\pm\,$’ is read as ‘plus or minus’ The plus sign (top left) in $\,\pm\,$ goes with the plus sign (top right) in $\,\pm\,.$ The minus sign (bottom left) in $\,\pm\,$ goes with the minus sign (bottom right) in $\,\pm\,.$ Subtraction Formula for Sine Difference Formula for Sine Sine Subtraction Formula Sine Difference Formula $$\sin(a\color{red}{\bf -}b) = \sin a\ \cos b \color{red}{\bf -} \cos a\ \sin b$$ Memory Device: For the sine sum/difference formulas: when there's a plus sign on the left, there's a plus sign on the right; when there's a minus sign on the left, there's a minus sign on the right. Thus, Sine is the Same. (For the cosine formula, they're different.)

## 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:

$$\cssId{s55}{\overbrace{\strut\cos(a + b)}^{\text{the cosine of a sum}}} \cssId{s56}{\overbrace{\strut=}^{\text{is}}} \cssId{s57}{\overbrace{\strut\cos a}^{\text{cosine of first }}} \cssId{s58}{\overbrace{\strut\cdot}^{\text{ times }}} \cssId{s59}{\overbrace{\strut\cos b}^{\text{ cosine of second }}} \cssId{s60}{\overbrace{\strut - }^{\text{ minus }}} \cssId{s61}{\overbrace{\strut\sin a}^{\text{ sine of first }}} \cssId{s62}{\overbrace{\strut\cdot}^{\text{ times }}} \cssId{s63}{\overbrace{\strut\sin b}^{\text{ sine of second}}}$$

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:

$$\cssId{s74}{\overbrace{\strut\sin(a + b)}^{\text{the sine of a sum}}} \cssId{s75}{\overbrace{\strut=}^{\text{is}}} \cssId{s76}{\overbrace{\strut\sin a}^{\text{sine of first }}} \cssId{s77}{\overbrace{\strut\cdot}^{\text{ times }}} \cssId{s78}{\overbrace{\strut\cos b}^{\text{ cosine of second }}} \cssId{s79}{\overbrace{\strut + }^{\text{ plus }}} \cssId{s80}{\overbrace{\strut\cos a}^{\text{ cosine of first }}} \cssId{s81}{\overbrace{\strut\cdot}^{\text{ times }}} \cssId{s82}{\overbrace{\strut\sin b}^{\text{ sine of second}}}$$

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}

Master the ideas from this section