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Sum to product and product to sum

Derive sum to product and product to sum formulae from compound angles. Evaluate exact expressions, factor trigonometric equations, retain zero factors and check quotient domains.

Before you startCompound angle formulae, exact values and periodic root sets

01 / Use the average and half-difference of the angles

This turns two compound expansions into a factorisation.

Let M = (A + B)/2 and D = (A − B)/2. Then A = M + D and B = M − D. Adding or subtracting the corresponding compound-angle expansions produces useful products.

M = (A + B)/2, D = (A − B)/2
A = M + D, B = M − D

The model links this algebra to a combined wave. Its product form explains why the curve stays between the blue envelopes and where its zero factors can occur.

A sum becomes a productExplore

ExpressionAngles A, BInput x

sin(5x) + sin(3x) = 2sin(4x)cos(x).

At x = 30°: A = 150°, B = 90°; M = 120°, D = 30°.

Sum/difference = 1.5000. Product = 1.5000.

Gold is the combined wave. Blue dashed curves are the positive and negative product envelopes. The remaining sine or cosine factor stays between −1 and 1. Decimal readouts are rounded to four places.

02 / Add and subtract the sine expansions

One pair of terms cancels in each case.

Expand sin(M + D) = sin M cos D + cos M sin D and sin(M − D) = sin M cos D − cos M sin D.

sin A + sin B = 2sin((A + B)/2)cos((A − B)/2)
sin A − sin B = 2cos((A + B)/2)sin((A − B)/2)

In the sum, the cosine-sine terms cancel. In the difference, the sine-cosine terms cancel. These identities hold for every real A and B.

Watch the combined wave stay inside its product envelope

Pause, replay or seek freely. The notes explain the same idea and stay in view.

03 / Subtract cosine expansions carefully

The cosine-difference formula carries a minus sign.

Expand cos(M + D) = cos M cos D − sin M sin D and cos(M − D) = cos M cos D + sin M sin D.

cos A + cos B = 2cos((A + B)/2)cos((A − B)/2)
cos A − cos B = −2sin((A + B)/2)sin((A − B)/2)

Keep the order A − B consistent. If you reverse the half-difference, its sine changes sign and the leading minus must change too.

04 / Read the identities backwards to turn products into sums

Take A = u + v and B = u − v.

2sin u cos v = sin(u + v) + sin(u − v)
2cos u cos v = cos(u + v) + cos(u − v)
2sin u sin v = cos(u − v) − cos(u + v)

Rewrite sin 5x cos 2x and sin 5x sin 2x as sums.Worked example

sin 5x cos 2x = (sin 7x + sin 3x)/2

Divide the first product identity by 2.

sin 5x sin 2x = (cos 3x − cos 7x)/2

The difference-angle cosine comes first.

For 2cos u sin v, commute the two factors and use the first identity with their roles swapped. This gives sin(u + v) − sin(u − v).

05 / Choose sums with familiar averages and half-differences

Exact values can emerge without evaluating each term.

Find sin 75° + sin 15° and cos 75° − cos 15°.Worked example

The average is 45° and the half-difference is 30°

Compute these before selecting a formula.

sin 75° + sin 15° = 2sin 45° cos 30° = √6/2

The sum uses sine of the average.

cos 75° − cos 15° = −2sin 45° sin 30° = −√2/2

The negative sign is essential.

06 / Factor a sum before solving it

Either zero factor can create a solution.

Solve sin 4x + sin 2x = 0 for 0° ≤ x ≤ 360°.Worked example

sin 4x + sin 2x = 2sin 3x cos x

Use average 3x and half-difference x.

sin 3x = 0 or cos x = 0

Do not divide by either trig factor.

x = 0°, 60°, 120°, 180°, 240°, 300°, 360° from sine

List every multiple-angle root in the interval.

x = 90°, 270° from cosine

These give two additional roots.

Combine the two lists

The complete set has nine distinct solutions.

07 / Merge roots shared by two factors

A repeated route to a root does not create a second angle.

Solve sin 5x + sin 3x = 0 on 0° ≤ x ≤ 360°.Worked example

2sin 4x cos x = 0

Factor using the sum formula.

sin 4x = 0 gives x = 45°k

Take k = 0, 1, …, 8.

cos x = 0 gives x = 90°, 270°

Both already belong to the first list.

The complete set is 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°

There are nine roots, not eleven.

It would still be invalid to divide by cos x without checking its zeros separately, even though in this particular example those roots happen to be recovered by the other factor.

08 / Use a cosine difference to reveal two sine factors

The leading nonzero constant does not affect the zeros.

Solve cos 5x − cos x = 0 on 0° ≤ x ≤ 360°.Worked example

cos 5x − cos x = −2sin 3x sin 2x

The average is 3x and the half-difference 2x.

sin 3x = 0 or sin 2x = 0

The constant −2 is never zero.

The first family is x = 60°k; the second is x = 90°k

Filter both into the required interval.

x = 0°, 60°, 90°, 120°, 180°, 240°, 270°, 300°, 360°

Merge shared endpoints and the shared root 180°.

09 / After cancellation, keep the original denominator restrictions

Equal simplified formulas can have different domains.

Simplify (sin 5x + sin 3x)/(cos 5x + cos 3x).Worked example

The numerator is 2sin 4x cos x

Factor the sine sum.

The denominator is 2cos 4x cos x

The original domain requires cos 4x cos x ≠ 0.

The quotient is tan 4x on that domain

Disprove cos A − cos B = 2sin((A + B)/2)sin((A − B)/2) using A = 60°, B = 0°.

Hint

Compare the two exact values.

Worked solution

The left side is 1/2 − 1 = −1/2; the proposed right side is 2 × 1/2 × 1/2 = 1/2. The correct formula has a leading minus.

11 / Recap

Average, half-difference, then factor.

Derive the identities by adding or subtracting compound-angle expansions.
Keep the minus sign in the cosine-difference formula.
Reverse the identities to convert products into sums.
For zero-product equations, solve every factor and merge duplicate roots.
After simplifying a quotient, retain the original denominator restrictions.

Section 1 of 11 · Use the average and half-difference of the angles