Can I use Sum to Product on \sin A + \cos B?

No, Sum to Product requires both terms to be the same trigonometric function (both sines or both cosines). If you have mixed functions, use a cofunction identity first to convert one term (e.g., \cos B = \sin(90° - B)), then apply Sum to Product.

What if one of the angles is negative or greater than 360°?

Sum to Product works for any real angles, including negative and large angles. Treat negative angles normally (they represent rotation in the opposite direction), compute \frac{A+B}{2} and \frac{A-B}{2} as usual, and proceed with the formula—the result will be correct regardless.

Does Sum to Product work for \sin A - \sin B or \cos A - \cos B?

Yes, but the formulas differ: \sin A - \sin B = 2\cos\frac{A+B}{2} \sin\frac{A-B}{2} and \cos A - \cos B = -2\sin\frac{A+B}{2} \sin\frac{A-B}{2}. Note the swapped roles of sine and cosine, and the negative sign in the cosine difference formula.

Why not just use a calculator to add the sines directly?

A calculator gives a decimal approximation, which is useful for numerical evaluation, but Sum to Product reveals the exact algebraic structure and relationships. This structure is essential for solving equations, proving identities, and understanding how two wave-like quantities combine mathematically—insights that a calculator cannot provide.

Sum to Product Formulas

Convert sums of sines or cosines into products, making trigonometric equations and expressions easier to simplify and solve.

\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}

Solve a problem with Sum to Product Formulas

Type the problem. The solver will use Sum to Product Formulas where Sum to Product Formulas is the right tool, and tell you when it is not.

What each symbol means

What Sum to Product Formulas takes

Symbol	Meaning
$A$	The first angle in the sum, measured in radians or degrees; confusing it with a coefficient or constant multiplier will lead to applying the formula incorrectly.
$B$	The second angle in the sum, measured in radians or degrees; reversing the order of $A$ and $B$ changes the signs and phase shift in the resulting product.

When to use it

Reach for Sum to Product Formulas when you see a sum or difference of two sine or cosine terms that you need to factor or simplify.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Simple sum of sines with known angles

Problem

Simplify

\sin 60° + \sin 30°

using Sum to Product Formulas.

$\frac{A+B}{2} = \frac{60° + 30°}{2} = 45°$
We identify $A = 60°$ and $B = 30°$ , then calculate the average of the two angles.
$\frac{A-B}{2} = \frac{60° - 30°}{2} = 15°$
We calculate the half-difference of the angles.
$\sin 60° + \sin 30° = 2\sin 45° \cos 15°$
We apply the Sum to Product formula directly with our calculated values.

Answer: $2\sin 45° \cos 15°$

This example shows a straightforward application of the formula. By recognizing $A$ and $B$ from the original sum, we convert two separate sine values into a product form that reveals the underlying structure—a product of a sine at the average angle and a cosine at the half-difference.

2. Simplifying a sum with variable angles

Problem

Simplify

\sin(x + \frac{\pi}{3}) + \sin(x - \frac{\pi}{3})

using Sum to Product Formulas.

$\frac{A+B}{2} = \frac{(x + \frac{\pi}{3}) + (x - \frac{\pi}{3})}{2} = \frac{2x}{2} = x$
We set $A = x + \frac{\pi}{3}$ and $B = x - \frac{\pi}{3}$ , then calculate the average; the $\frac{\pi}{3}$ terms cancel.
$\frac{A-B}{2} = \frac{(x + \frac{\pi}{3}) - (x - \frac{\pi}{3})}{2} = \frac{\frac{2\pi}{3}}{2} = \frac{\pi}{3}$
We calculate the half-difference; the $x$ terms cancel and we're left with $\frac{\pi}{3}$ .
$\sin(x + \frac{\pi}{3}) + \sin(x - \frac{\pi}{3}) = 2\sin x \cos \frac{\pi}{3}$
We apply the Sum to Product formula with our calculated average and half-difference.
$2\sin x \cdot \frac{1}{2} = \sin x$
We substitute the known value $\cos \frac{\pi}{3} = \frac{1}{2}$ and simplify.

Answer: $\sin x$

This example is crucial because it shows Sum to Product's power with variables. When angles are symmetric around a central value, the formula produces a remarkably clean result—here, a simple expression in $x$ alone. This demonstrates why Sum to Product is so useful in solving trigonometric equations.

3. Combining signals in a practical context

Problem

A bake sale organizer uses two oscillating display lights. Light 1's brightness follows

\sin 70°

and light 2 follows

\sin 20°

. Find the combined brightness

\sin 70° + \sin 20°

using Sum to Product Formulas to understand how they reinforce each other.

$\frac{A+B}{2} = \frac{70° + 20°}{2} = 45°$
We identify $A = 70°$ and $B = 20°$ , then compute the average angle.
$\frac{A-B}{2} = \frac{70° - 20°}{2} = 25°$
We compute the half-difference angle.
$\sin 70° + \sin 20° = 2\sin 45° \cos 25°$
We apply the Sum to Product formula to convert the sum of brightness values into a product.
$2 \cdot \frac{\sqrt{2}}{2} \cdot \cos 25° = \sqrt{2} \cos 25°$
We substitute $\sin 45° = \frac{\sqrt{2}}{2}$ to express the result in simplest form.

Answer: $\sqrt{2} \cos 25°$

This practical example reveals how two signals combine in the real world. Sum to Product shows that the combined brightness is proportional to $\cos 25°$ , with $\sqrt{2}$ as the amplitude factor. This structure is exactly what engineers and physicists use to analyze wave interference and signal addition.

Common mistakes

Where Sum to Product Formulas usually goes wrong

The mistake	Why it is wrong	The fix
$\sin A + \sin B = \sin\frac{A+B}{2} \cos\frac{A-B}{2}$ (forgetting the factor of 2)	The factor of 2 is essential to the identity; without it, the right side equals only half the value of the left side, making the equation false.	Always include the 2 in front: $\sin A + \sin B = 2\sin\frac{A+B}{2} \cos\frac{A-B}{2}$ .
$\sin A + \sin B = 2\sin\frac{A-B}{2} \cos\frac{A+B}{2}$ (swapping the average and half-difference in the argument positions)	The positions matter: sine must pair with the average angle $\frac{A+B}{2}$ , and cosine with the half-difference $\frac{A-B}{2}$ . Swapping them produces an incorrect identity that doesn't equal the original sum.	Remember the pattern: $\sin$ gets $\frac{A+B}{2}$ (the average), and $\cos$ gets $\frac{A-B}{2}$ (the half-difference).
Attempting to use Sum to Product on $\sin A \cos B$ or $2\sin A \cos B$ without recognizing it is already a product	Sum to Product converts a sum into a product; if you already have a product, you need the inverse transformation (product-to-sum formulas), and applying the wrong formula wastes effort and creates errors.	Before applying Sum to Product, verify you have a sum or difference of the same trig function (e.g., $\sin A + \sin B$ or $\cos A - \cos B$ ), not a product.

Tips and when to use something else

Sum to Product formulas have an inverse relationship with product-to-sum formulas (e.g., $2\sin X \cos Y = \sin(X+Y) + \sin(X-Y)$ ); if you get stuck solving an equation, try working backwards with product-to-sum instead.
The average angle $\frac{A+B}{2}$ typically dominates the result's behavior (it carries the main frequency or trend), while the half-difference $\frac{A-B}{2}$ acts as an envelope or modulation factor that adjusts the amplitude.
For a difference formula, $\sin A - \sin B = 2\cos\frac{A+B}{2} \sin\frac{A-B}{2}$ , the roles of sine and cosine are swapped; this is a different identity, so memorize both the sum and difference versions.
If neither $\frac{A+B}{2}$ nor $\frac{A-B}{2}$ simplifies to a known angle or nice variable, consider using the Sine or Cosine Addition Formulas as an alternative approach to expand and recombine the terms.

Related formulas

Sine Addition FormulaThe Sine Addition Formula expands sine of a sum or difference into a product of sines and cosines, making it useful for exact angle calculations.Cosine Addition FormulaFind the cosine of an angle sum or difference by using the cosine and sine values of the individual angles—essential for solving trigonometric equations.Double Angle FormulasDouble angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.Solving Trigonometric EquationsFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.

Frequently asked questions

Can I use Sum to Product on $\sin A + \cos B$ ?: No, Sum to Product requires both terms to be the same trigonometric function (both sines or both cosines). If you have mixed functions, use a cofunction identity first to convert one term (e.g., $\cos B = \sin(90° - B)$ ), then apply Sum to Product.
What if one of the angles is negative or greater than $360°$ ?: Sum to Product works for any real angles, including negative and large angles. Treat negative angles normally (they represent rotation in the opposite direction), compute $\frac{A+B}{2}$ and $\frac{A-B}{2}$ as usual, and proceed with the formula—the result will be correct regardless.
Does Sum to Product work for $\sin A - \sin B$ or $\cos A - \cos B$ ?: Yes, but the formulas differ: $\sin A - \sin B = 2\cos\frac{A+B}{2} \sin\frac{A-B}{2}$ and $\cos A - \cos B = -2\sin\frac{A+B}{2} \sin\frac{A-B}{2}$ . Note the swapped roles of sine and cosine, and the negative sign in the cosine difference formula.
Why not just use a calculator to add the sines directly?: A calculator gives a decimal approximation, which is useful for numerical evaluation, but Sum to Product reveals the exact algebraic structure and relationships. This structure is essential for solving equations, proving identities, and understanding how two wave-like quantities combine mathematically—insights that a calculator cannot provide.

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Reviewed 2026-09-18