Addition and Subtraction Formulas for Sine and Cosine


LESSON READ-THROUGH
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’,
instead of ‘sine of ex’.
 

This section explores four trigonometric identities:

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 $\,\cos(a + b)\,$:



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 $\,\sin(a + b)\,$:

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
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Double Angle Formulas for Sine and Cosine

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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