How can I understand and prove the "sum and difference formulas" in trigonometry?

Question

The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.

\begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}

So what I want to know is,

1. How can I prove that these formulas are correct?
2. More importantly, how can I understand these formulas intuitively?

Ideally, I'm looking for answers that make no reference to Calculus, or to Euler's formula, although such answers are still encouraged, for completeness.

Answer 1

The key fact here is that rotation is a linear transformation, e.g. the rotation of $u + v$ is the rotation of $u$ plus the rotation of $v$. You should draw a diagram that shows this carefully if you don't believe it. That means a rotation is determined by what it does to $(1, 0)$ and to $(0, 1)$.

But $(1, 0)$ rotated by $\theta$ degrees counterclockwise is just $(\cos \theta, \sin \theta)$, whereas $(0, 1)$ rotated by $\theta$ degrees counterclockwise is just $(-\sin \theta, \cos \theta)$. (Again, draw a diagram.) That means a rotation by $\theta$ is given by a $2 \times 2$ matrix with those entries. (Matrices don't work here yet.)

So take a rotation by $\theta$ and another one by $\theta'$, and multiply the corresponding matrices. What you get is the sine and cosine angle addition formulas. (The connection to complex numbers is that one can represent complex numbers as $2 \times 2$ real matrices.)

Also, if you believe that $a \cdot b = |a| |b| \cos \theta$, this implies the cosine angle difference formula when $a$ and $b$ are unit vectors. Ditto for the cross product and the sine angle difference formula.

Answer 2

There are several typical derivations used in high school texts. Here's one:

diagram http://www.imgftw.net/img/400545892.png

Take two points on the unit circle, one a rotation of (1,0) by α, the other a rotation of (1,0) by β. Their coordinates are as shown in the diagram. Let c be the length of the segment joining those two points. By the Law of Cosines (on the blue triangle), $c^2=1^2+1^2-2\cdot1\cdot1\cdot\cos(\alpha-\beta)$. Using the distance formula, $c=\sqrt{(\cos\alpha-\cos\beta)^2+(\sin\alpha-\sin\beta)^2}$. Squaring the latter and setting the two equal, $1^2+1^2-2\cdot1\cdot1\cdot\cos(\alpha-\beta)=(\cos\alpha-\cos\beta)^2+(\sin\alpha-\sin\beta)^2$. Simplifying both sides, $2-2\cos(\alpha-\beta)=\cos^2\alpha-2\cos\alpha\cos\beta+\cos^2\beta+\sin^2\alpha-2\sin\alpha\sin\beta+\sin^2\beta$ $=2-2\cos\alpha\cos\beta-2\sin\alpha\sin\beta$ (using the Pythagorean identity). Solving for $\cos(\alpha-\beta)$, $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$.

From this identity, the other three can be derived by substituting $\frac{\pi}{2}-\alpha$ for α (gives sin(α+β)), then -β for β (gives the remaining two).

As to understanding the formulas intuitively if you accept that multiplying by a complex number $z_\theta$ for which |z|=1 rotates by θ, then you can think about what happens when you multiply $z_\alpha=\cos\alpha+i\sin\alpha$ and $z_\beta=\cos\beta+i\sin\beta$ (by expanding the binomial product), which should result in $\cos(\alpha+\beta)+i\sin(\alpha+\beta)$.

Answer 3

You can use the complex representation,
$\cos(x) = \frac{1}{2}(e^{ix} + e^{-ix})$
$\sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$
and the rules for powers ($a^{x+y}=a^x a^y$)

Answer 4

Though the standard high-school derivations are not the most useful way to remember it in the long run, here's another one which I like because you can "see" it directly without much algebra.

Let P be the point on the unit circle got by rotating (1,0) by angle α+β. Drop a perpendicular N to the α-rotated line, and R to the x-axis. So from the right triangle ONP, you see ON = cos β. You can see that the angle RPN is α too: it's the complement of ∠PNQ, and so is ∠QNO = α. Now,

$\sin(\alpha + \beta) = \mbox{PR} = \mbox{PQ} + \mbox{QR} = \sin(\beta)\cos(\alpha) + \cos(\beta)\sin(\alpha)$, and

$\cos(\alpha + \beta) = \mbox{OR} = \mbox{OM} - \mbox{RM} = \cos(\beta)\cos(\alpha) - \sin(\beta)\sin(\alpha)$.

Answer 5

I remember that $e^{i\alpha}=\cos\alpha+i\sin\alpha$ and that $i^2=-1$. Both these relations are useful in many other situations and pretty fundamental to understanding complex numbers. Then your equalities are the real and, respectively, the imaginary part of $e^{i(\alpha+\beta)}=e^{i\alpha}e^{i\beta}$.

This is not very different from the other answers, but I actually prefer the algebra perspective. The only place where I think geometrically is in interpreting $e^{i\alpha}=\cos\alpha+i\sin\alpha$ by thinking of the unit circle in the complex plane.

Answer 6

Here are my favorite diagrams:

As given, the diagrams put certain restrictions on the angles involved: neither angle, nor their sum, can be larger than 90 degrees; and neither angle, nor their difference, can be negative. The diagrams can be adjusted, however, to push beyond these limits. (See, for instance, this answer.)

Here's a bonus mnemonic cheer (which probably isn't as exciting to read as to hear):

Sine, Cosine, Sign, Cosine, Sine!
Cosine, Cosine, Co-Sign, Sine, Sine!

The first line encapsulates the sine formulas; the second, cosine. Just drop the angles in (in order $\alpha$, $\beta$, $\alpha$, $\beta$ in each line), and know that "Sign" means to use the same sign as in the compound argument ("+" for angle sum, "-" for angle difference), while "Co-Sign" means to use the opposite sign.

Answer 7

Consider a unit circle with $O$ as the centre. Let $P_{1}$, $P_2$ and $P_{3}$ be points on the circle making angles $A$, $B$ and $A−B$, respectively, with the positive direction of the X-axis. We know that if two chords subtend equal angle at the centre, then the chords are equal and chords $P_{3}P_{0}$ and $P_1P_2$ subtend equal angles at $O$. Therefore, $\overline{P_{3}P_{0}}=\overline{P_1P_2}$. By distance formula, the distance between the points $P_{0}(1,0)$ and $P_{3}(\cos(A−B),\sin(A−B))$ is $$P_{3}P_{0}=\sqrt{\left(\cos \left( A-B \right) - 1\right) ^{2} + \left(\sin \left( A-B \right) - 0\right) ^{2}}$$ Similarly, the distance between the points $P_{1}(\cos A,\sin A)$ and $P_{2}(\cos B,\sin B)$ is $$P_1P_2= \sqrt{\left( \cos B-\cos A\right) ^{2}+\left( \sin B - \sin A \right) ^{2}}$$

On squaring both sides, we get

$$ \begin{array}{ll} \Rightarrow & \left\{\cos (A-B)-\left.1\right|^{2}+\sin ^{2}(A-B)=(\cos B-\cos A)^{2}+(\sin B-\sin A)^{2}\right. \\ \Rightarrow & \cos ^{2}(A-B)-2 \cos (A-B)+1+\sin ^{2}(A-B)=\cos ^{2} B+\cos ^{2} A-2 \cos A \cos B \\ & +\sin ^{2} B+\sin ^{2} A-2 \sin A \sin B \\ \Rightarrow & 2-2 \cos (A-B)=2-2 \cos A \cos B-2 \sin A \sin B \\ \Rightarrow & \cos (A-B)=\cos A \cos B+\sin A \sin B \\ \text { Hence, } & \cos (A-B)=\cos A \cos B+\sin A \sin B \end{array} $$

Answer 8

Let $\alpha, \beta, \theta \in \mathbb{R}.$ Consider vectors $\vec{P} = (\cos(\alpha), \sin(\alpha)), \vec{Q} = (\cos(\beta), \sin(\beta) ),$ and their rotated versions $\vec{P'} = (\cos(\alpha + \theta), \sin(\alpha + \theta) ), \vec{Q'} = (\cos(\beta + \theta), \sin(\beta + \theta) ).$

Rotations preserve distances, so $PQ = P'Q',$ ie $(\cos(\alpha) - \cos(\beta)) ^2 + (\sin(\alpha) - \sin(\beta) ) ^2$ $= (\cos(\alpha + \theta) - \cos(\beta + \theta) ) ^2 + (\sin(\alpha + \theta) - \sin(\beta + \theta) ) ^2 ,$ ie ${\color{green}{\cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta) = \cos(\alpha + \theta) \cos(\beta + \theta) + \sin(\alpha + \theta) \sin(\beta + \theta)}}.$

This holds for all values of $\alpha, \beta, \theta.$
Setting $\alpha = 0,$ $\cos(\beta) = \cos(\theta) \cos(\beta + \theta) + \sin(\theta) \sin(\theta + \beta).$ Further setting $\lbrace \beta = A+B, \theta = (-B) \rbrace$ gives $\cos(A+B) = \cos(A) \cos(B) - \sin(A) \sin(B).$

Similarly setting $\alpha = \frac{\pi}{2}$ gives $\sin(\beta) = -\sin(\theta) \cos(\beta + \theta) + \cos(\theta) \sin(\beta + \theta),$ and further setting ${ \lbrace \beta = A + B, \theta = (-B) \rbrace }$ gives ${ \sin(A+B) = \sin(A) \cos(B) + \cos(A) \sin(B) }.$

Answer 9

The following is (essentially) a graphical version of an earlier answer, relying on some simple vector operations in two dimensions but without explicitly invoking matrices.

The argument is summarized in the following diagram.

This is meant to be a proof without words, but for clarity, I'll build up to the conclusion step by step below.

First, let's just consider the unit circle in the usual Cartesian plane. Here it is, along with the $x$ and $y$ axes and unit vectors from the origin to the two points $(1,0)$ and $(0,1).$ We use $A=(1,0)$ and $B=(0,1)$ to denote the position vectors of these two points.

Next we add a point at an angle $\alpha$ counterclockwise around the circle from the point $A=(1,0).$ According to the unit circle interpretation of the sine and cosine, the coordinates of this point, and hence the coordinates of its position vector, are $C = (\cos\alpha, \sin\alpha).$

But as illustrated in by the vector sum figure, this is equivalent to saying that $C$ is a linear combination of the two vectors $A=(1,0)$ and $B=(0,1),$

$$ C = (\cos\alpha) A + (\sin\alpha) B. $$

The equation above is a coordinate-free way of constructing a unit position vector rotated an angle $\theta$ counterclockwise from a given unit vector: find a unit vector at a right angle counterclockwise from the given vector, and the rotated vector is $\cos\theta$ of the given vector combined with $\sin\theta$ of the right-angle vector.

Now that we have the unit vector $C$, we construct a unit vector $D$ at a right angle clockwise from $C$:

The vector sum figure, rotated $90$ degrees from the figure in the previous diagram, shows that the $x$ coordinate of $D$ is the opposite of the $y$ coordinates of $C$ while the $y$ coordinate of $D$ is simply the $x$ coordinates of $C.$ If this is not obvious enough, you can verify it for a vector $C$ in each of the four quadrants.

Now we construct a unit vector $E$ at an angle $\beta$ counterclockwise from $C.$

We construct this vector as a linear combination of the given vector $C$ and the vector $D$ at a right angle counterclockwise from $C,$ that is,

$$ E = (\cos\beta) C + (\sin\beta) D. $$

Finally, we multiply the coordinates of $C$ by the scalar $\cos\beta$, multiply the coordinates of $D$ by the scalar $\sin\beta$, and add the resulting vectors coordinatewise to get the coordinates of $E$:

$$ E = (\cos\beta\cos\alpha - \sin\beta\sin\alpha, \cos\beta\sin\alpha + \sin\beta\cos\alpha). $$

But $E$ is a vector at an angle $\alpha + \beta$ counterclockwise from $A=(1,0),$ and therefore its coordinates give two trigonometric functions of $\alpha + \beta,$

\begin{align} \cos(\alpha + \beta) &= \cos\beta\cos\alpha - \sin\beta\sin\alpha, \\ \sin(\alpha + \beta) &= \cos\beta\sin\alpha + \sin\beta\cos\alpha. \end{align}

Simple rearrangement within the terms gets us the usual formulas for these functions.

Answer 10

There are simple geometric proofs of the formulas for $\sin(\alpha \pm \beta)$ and $\cos(\alpha \pm \beta)$ for the case where $\alpha,$ $\beta,$ and $\alpha \pm \beta$ are all acute angles. The answer I am linking here is a great example.

To move beyond acute angles you need to have a notion of what the sines and cosines of non-acute angles are. One can extend the graphical proofs to other cases one by one, but wouldn't it be nice to cover all possible angles once and for all? So let's use a few additional facts that can be more or less read directly from the unit circle definition, for example $\sin(\tfrac\pi2 - \theta) = \cos(\theta)$ and $\sin(\pi - \theta) = \sin(\theta).$

If $\alpha + \beta = \tfrac\pi2$ then $\beta = \tfrac\pi2 - \alpha,$ \begin{align} \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) &= \sin(\alpha)\cos(\tfrac\pi2 - \alpha) + \cos(\alpha)\sin(\tfrac\pi2 - \alpha) \\ &= \sin^2(\alpha) + \cos^2(\alpha) \\ &= 1 = \sin(\alpha + \beta), \end{align} and \begin{align} \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) &= \cos(\alpha)\cos(\tfrac\pi2 - \alpha) - \sin(\alpha)\sin(\tfrac\pi2 - \alpha) \\ &= \cos(\alpha)\sin(\alpha) - \sin(\alpha)\cos(\alpha) \\ &= 0 = \sin(\alpha + \beta), \end{align} so the sum formulas both work in that case.

If $\alpha + \beta$ is an obtuse angle then \begin{align} \sin(\alpha)\cos&(\beta) + \cos(\alpha)\sin(\beta) \\ &= \cos(\tfrac\pi2 - \alpha)\sin(\tfrac\pi2 - \beta) + \sin(\tfrac\pi2 - \alpha)\cos(\tfrac\pi2 - \beta) \\ &= \sin((\tfrac\pi2 - \alpha) + (\tfrac\pi2 - \beta)) \\ &= \sin(\pi - (\alpha + \beta)) \\ &= \sin(\alpha + \beta) \\ \end{align} and \begin{align} \cos(\alpha)\cos&(\beta) - \sin(\alpha)\sin(\beta) \\ &= \sin(\tfrac\pi2 - \alpha)\sin(\tfrac\pi2 - \beta) - \cos(\tfrac\pi2 - \alpha)\cos(\tfrac\pi2 - \beta) \\ &= -\cos((\tfrac\pi2 - \alpha) + (\tfrac\pi2 - \beta)) \\ &= -\cos(\pi - (\alpha + \beta)) \\ &= \cos(\alpha + \beta), \\ \end{align} so both sum formulas work in that case too.

There are no other cases for the sum of two acute angles, but let's also consider the zero angle. If $\alpha = 0$ then \begin{align} \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) &= 0 \cdot \cos(\beta) + 1\cdot \sin(\beta) \\ &= \sin(\beta) \\ &= \sin(\alpha + \beta) \end{align} and \begin{align} \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) &= 1 \cdot \cos(\beta) + 0\cdot \sin(\beta) \\ &= \cos(\beta) \\ &= \cos(\alpha + \beta), \end{align} so the formulas work in that case, and by symmetry they also work when $\beta = 0$. Obviously they work when both angles are zero.

That covers all possibilities for the sum of angles $\alpha$ and $\beta$ where $0 \leq \alpha \lt \tfrac\pi2$ and $0 \leq \beta \lt \tfrac\pi2.$

To move beyond this, let me define an ad-hoc notation to help express some formulas more conveniently. The ad-hoc notation makes the formulas less cluttered and (I hope) easier to grasp. For arbitrary angles $\theta,$ define $$ \sin_k(\theta) \triangleq \sin(\theta + k \cdot \tfrac\pi2). $$

Then \begin{align} \sin_0(\theta) &= \sin(\theta + 0 \cdot \tfrac\pi2) = \sin(\theta), \\ \sin_1(\theta) &= \sin(\theta + 1 \cdot \tfrac\pi2) = \cos(\theta), \\ \sin_2(\theta) &= \sin(\theta + 2 \cdot \tfrac\pi2) = -\sin(\theta), \\ \sin_3(\theta) &= \sin(\theta + 3 \cdot \tfrac\pi2) = -\cos(\theta) \\ \end{align} for any angle $\theta$ (not just acute angles), as one can easily verify using the unit circle, and the pattern repeats four lines at a time (either after or before this set of equations) by adding or subtracting $2\pi$ from each angle; $\sin_{k+4}(\theta) = \sin_k(\theta) = \sin_{k-4}(\theta).$ Also note that $\sin_{k+2}(\theta) = -\sin_k(\theta) = \sin_{k-2}(\theta).$

Then for any integers $m$ and $n,$ \begin{align} \sin_m(\alpha)\sin_{n+1}(\beta) &= (-\sin_{m+2}(\alpha))(-\sin_{n-1}(\beta)) \\ &= \sin_{m+2}(\alpha)\sin_{n-1}(\beta). \end{align}

It follows that \begin{align} \sin_m(\alpha)\sin_{n+1}&(\beta) + \sin_{m+1}(\alpha)\sin_n(\beta) \\ &= \sin_{m+1}(\alpha)\sin_n(\beta) + \sin_{m+2}(\alpha)\sin_{n-1}(\beta) \end{align} by increasing $m$ by $2$ and decreasing $n+1$ by $2$ in $\sin_m(\alpha)\sin_{n+1}(\beta)$ and then reversing the order of the sum. The result is that the "$m$" subscripts both increase by $1$ and the "$n$" subscripts both decrease by $1.$ We can repeat this set of operations as many times as we like, forward or backward, so for any integer $k,$ \begin{align} \sin_m(\alpha)\sin_{n+1}&(\beta) + \sin_{m+1}(\alpha)\sin_n(\beta) \\ &= \sin_{m+k}(\alpha)\sin_{n-k+1}(\beta) + \sin_{m+k+1}(\alpha)\sin_{n-k}(\beta). \end{align}

In particular, for $k = n,$ \begin{align} \sin_m(\alpha)\sin_{n+1}&(\beta) + \sin_{m+1}(\alpha)\sin_n(\beta) \\ &= \sin_{m+n}(\alpha)\sin_1(\beta) + \sin_{m+n+1}(\alpha)\sin_0(\beta). \end{align}

Now that we have all this machinery, let $\alpha$ and $\beta$ be any angles of any magnitudes, positive or negative. Write \begin{align} \alpha &= \bar\alpha + a\cdot \tfrac\pi2, \\ \beta &= \bar\beta + b\cdot \tfrac\pi2. \end{align}

Then $$\sin(\alpha + \beta) = \sin(\bar\alpha + \bar\beta + (a + b)\tfrac\pi2) = \sin_{a+b}(\bar\alpha + \bar\beta).$$ We can cover all possible values of $\alpha$ and $\beta$ in just four cases, depending on the equivalence class of $a + b$ modulo $4.$ In each case we can rely on the knowledge of the angle sum formulas for acute angles $\bar\alpha$ and $\bar\beta.$

Case $a + b \equiv 0 \pmod 4$:

\begin{align} \sin_{a+b}(\bar\alpha + \bar\beta) &= \sin(\bar\alpha + \bar\beta) = \sin(\bar\alpha)\cos(\bar\beta) + \cos(\bar\alpha)\sin(\bar\beta) \\ &= \sin_0(\bar\alpha)\sin_1(\bar\beta) + \sin_1(\bar\alpha)\sin_0(\bar\beta) \\ &= \sin_{a+b}(\bar\alpha)\sin_1(\bar\beta) + \sin_{a+b+1}(\bar\alpha)\sin_0(\bar\beta) \end{align}

Case $a + b \equiv 1 \pmod 4$:

\begin{align} \sin_{a+b}(\bar\alpha + \bar\beta) &= \cos(\bar\alpha + \bar\beta) = \cos(\bar\alpha)\cos(\bar\beta) - \sin(\bar\alpha)\sin(\bar\beta) \\ &= \sin_1(\bar\alpha)\sin_1(\bar\beta) + \sin_2(\bar\alpha)\sin_0(\bar\beta) \\ &= \sin_{a+b}(\bar\alpha)\sin_1(\bar\beta) + \sin_{a+b+1}(\bar\alpha)\sin_0(\bar\beta) \end{align}

Case $a + b \equiv 2 \pmod 4$:

\begin{align} \sin_{a+b}(\bar\alpha + \bar\beta) &= -\sin(\bar\alpha + \bar\beta) = -\sin(\bar\alpha)\cos(\bar\beta) - \cos(\bar\alpha)\sin(\bar\beta) \\ &= \sin_2(\bar\alpha)\sin_1(\bar\beta) + \sin_3(\bar\alpha)\sin_0(\bar\beta) \\ &= \sin_{a+b}(\bar\alpha)\sin_1(\bar\beta) + \sin_{a+b+1}(\bar\alpha)\sin_0(\bar\beta) \end{align}

Case $a + b \equiv 3 \pmod 4$:

\begin{align} \sin_{a+b}(\bar\alpha + \bar\beta) &= -\cos(\bar\alpha + \bar\beta) = -\cos(\bar\alpha)\cos(\bar\beta) + \sin(\bar\alpha)\sin(\bar\beta) \\ &= \sin_3(\bar\alpha)\sin_1(\bar\beta) + \sin_0(\bar\alpha)\sin_0(\bar\beta) \\ &= \sin_{a+b}(\bar\alpha)\sin_1(\bar\beta) + \sin_{a+b+1}(\bar\alpha)\sin_0(\bar\beta) \end{align}

Note that in all four cases we transformed $\sin_{a+b}(\bar\alpha + \bar\beta)$ to the same expression, so we can finish each case as follows: \begin{align} \sin_{a+b}(\bar\alpha)\sin_1&(\bar\beta) + \sin_{a+b+1}(\bar\alpha)\sin_0(\bar\beta) \\ &= \sin_a(\bar\alpha)\sin_{b+1}(\bar\beta) + \sin_{a+1}(\bar\alpha)\sin_b(\bar\beta) \\ &= \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \end{align}

Therefore $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$ for all angles $\alpha$ and $\beta.$

That's one of the four angle-sum/difference formulas for sine and cosine. The others follow easily now that we know that the formula for $\sin(\alpha + \beta)$ is not limited to positive acute angles. For $\sin(\alpha - \beta),$ \begin{align} \sin(\alpha - \beta) &= \sin(\alpha + (-\beta)) \\ &= \sin(\alpha)\cos(-\beta) + \cos(\alpha)\sin(-\beta) \\ &= \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta). \end{align}

For the cosine formulas, \begin{align} \cos(\alpha \pm \beta) &= \sin(\alpha \pm \beta + \tfrac\pi2) = \sin((\alpha + \tfrac\pi2) \pm \beta) \\ &= \sin(\alpha + \tfrac\pi2)\cos(\pm\beta) + \cos(\alpha + \tfrac\pi2)\sin(\pm\beta) \\ &= \cos(\alpha)\cos(\pm\beta) - \sin(\alpha)\sin(\pm\beta) \\ &= \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta). \end{align}

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That's not as many cases as one might have expected. I wonder if we can reduce it all down to one case by somehow observing that for acute angles $\bar\alpha$ and $\bar\beta,$ $$ \sin_k(\bar\alpha + \bar\beta) = \sin_k(\bar\alpha)\sin_1(\bar\beta) + \sin_{k+1}(\bar\alpha)\sin_0(\bar\beta), $$ but I have not hit on a method of proving that except by the four cases.