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:
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Subtraction Formula for Cosine Difference Formula for Cosine Cosine Subtraction Formula Cosine Difference Formula |
$$\cos(a\color{red}{\bf -}b) = \cos a\ \cos b \color{blue}{\bf +} \sin a\ \sin b$$
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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:
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Subtraction Formula for Sine Difference Formula for Sine Sine Subtraction Formula Sine Difference Formula |
$$\sin(a\color{red}{\bf -}b) = \sin a\ \cos b \color{red}{\bf -} \cos a\ \sin b$$ |
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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.) |
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 $\,\cos(a + b)\,$:
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 $\,\sin(a + b)\,$:
There are similar verbalizations and memory recall methods for the difference 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.
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.
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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.
Since subtraction is a special kind of addition, the difference formulas follow easily from the sum formulas. $$ \begin{alignat}{2} \cssId{s168}{\cos(a-b)}\ \ &\cssId{s169}{= \ \ \cos (a + (-b))} &\qquad&\cssId{s170}{\text{(to subtract $b$, add the opposite)}}\cr &\cssId{s171}{= \ \ \cos(a)\,\cos(-b) - \sin(a)\,\sin(-b)} &&\cssId{s172}{\text{(sum formula for cosine)}}\cr &\cssId{s173}{= \ \ \cos(a)\,\cos(b) - \sin(a)\bigl(-\sin(b)\bigr)}&&\cssId{s174}{\text{(cosine is even; sine is odd)}}\cr &\cssId{s175}{= \ \ \cos a\,\cos b + \sin a\,\sin b} &&\cssId{s176}{\text{(simplify)}}\cr\cr\cr\cr \cssId{s177}{\sin(a-b)}\ \ &\cssId{s178}{= \ \ \sin (a + (-b))} &\qquad&\cssId{s179}{\text{(to subtract $b$, add the opposite)}}\cr &\cssId{s180}{= \ \ \sin(a)\,\cos(-b) + \cos(a)\,\sin(-b)} &&\cssId{s181}{\text{(sum formula for sine)}}\cr &\cssId{s182}{= \ \ \sin(a)\,\cos(b) + \cos(a)\bigl(-\sin(b)\bigr)}&&\cssId{s183}{\text{(cosine is even; sine is odd)}}\cr &\cssId{s184}{= \ \ \sin a\,\cos b - \cos a\,\sin b} &&\cssId{s185}{\text{(simplify)}} \end{alignat} $$
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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