Sum-to-Product and Product-to-Sum

What this move does

Add the two unit vectors $u = (cos α, sin α)$ and $v = (cos β, sin β)$ and you get the diagonal of a rhombus. A diagonal of a rhombus bisects the angle at the vertex, so it points in the direction $\frac{α + β}{2}$ and has length $2 cos \frac{α - β}{2}$ . Compute the coordinates of this arrow once by “adding the pieces” and once as “length × direction”, and you get

cos α + cos β cos α - cos β = 2 cos \frac{α + β}{2} cos \frac{α - β}{2}, = - 2 sin \frac{α + β}{2} sin \frac{α - β}{2}, sin α + sin β sin α - sin β = 2 sin \frac{α + β}{2} cos \frac{α - β}{2}, = 2 cos \frac{α + β}{2} sin \frac{α - β}{2} .

Read from right to left, these are the product-to-sum formulas, for example $2 sin x cos y = sin (x + y) + sin (x - y)$ . Which direction to use when:

Two terms with the same function, added or subtracted: turn them into a product when you want to take out a common factor, decide a sign, cancel, or find zeros.
Two trig functions multiplied: turn them into a sum when you want to lower the power, or make a long expression cancel term by term.
The new angles are always the half-sum $\frac{α + β}{2}$ and the half-difference $\frac{α - β}{2}$ . If the functions differ, first use a reduction formula to make them the same.

Predict first

Which of these equals $cos 5 0^{\circ} + cos 7 0^{\circ}$ ?

cos 1 0^{\circ}

cos 12 0^{\circ}

3 cos 1 0^{\circ}

2 cos 6 0^{\circ} cos 2 0^{\circ}

Choose an answer first; then you can see the result.

Work through one

One diagonal, read two ways

Step 1 / 6

Two unit vectors
$u$ has direction angle $α = 8 0^{\circ}$ and $v$ has direction angle $β = 2 0^{\circ}$ . Their coordinates are just cosines and sines.

$u = (cos α, sin α), v = (cos β, sin β)$
The diagonal of a rhombus
How long is the diagonal?
Project onto the x-axis: read it twice
Project onto the y-axis
The other diagonal: the difference

Product-to-sum at its best: making a chain of terms cancel

Find $S = cos θ + cos 3 θ + cos 5 θ + \dots + cos (2 n - 1) θ$ . Multiply every term by $2 sin θ$ and use product-to-sum:

2 sin θ cos (2 k - 1) θ = sin 2 k θ - sin (2 k - 2) θ .

On the right, neighboring terms link up end to end, and when you add them only the two ends survive: $2 sin θ \cdot S = sin 2 n θ$ . This is what telescoping looks like in trigonometry. The factor $2 sin θ$ is chosen on purpose, so that after product-to-sum the angles shift by exactly one notch.

When it fails: different coefficients

Break it

Counterexample lab: remove one condition

Conclusion: sum-to-product writes $a cos α + b cos β$ as a single product

With both coefficients $1$ , the two vectors are equally long, the parallelogram is a rhombus, and the diagonal lies exactly on the angle bisector.

Practice

Exercise 1Basic

Find $sin 1 0^{\circ} + sin 5 0^{\circ} - sin 7 0^{\circ}$ .

Exercise 2Basic

Find $\frac{sin 7 5 ^{\circ} + sin 1 5 ^{\circ}}{cos 7 5 ^{\circ} + cos 1 5 ^{\circ}}$ .

Exercise 3Advanced

(A classic.) Find $sin 1 0^{\circ} sin 5 0^{\circ} sin 7 0^{\circ}$ .

Exercise 4Challenge

Find $cos \frac{π}{11} + cos \frac{3 π}{11} + cos \frac{5 π}{11} + cos \frac{7 π}{11} + cos \frac{9 π}{11}$ .

What this move does

Work through one

One diagonal, read two ways

Product-to-sum at its best: making a chain of terms cancel

When it fails: different coefficients

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Dropping the minus sign in cos α − cos β

Turning a sum into a product when the functions differ