4. Product and Sum Identities

Product and Sum Identities

Product-to-sum and sum-to-product identities are trigonometric relationships that allow us to rewrite products of sine and cosine functions as sums or differences, and vice versa.

For example, the product:

sin A cos B

can be rewritten as:

[sin(A + B) + sin(A − B)]/2

Similarly, the sum:

sin A + sin B

can be rewritten as:

2sin[(A + B)/2]cos[(A − B)/2]

These transformations are particularly useful when simplifying complicated expressions, solving trigonometric equations, analysing waves, and working with trigonometric integrals.

The identities are closely related to compound-angle identities, so understanding how they are derived makes them easier to remember and apply.

Understanding Products and Sums in Trigonometry

A product is the result of multiplying two expressions.

Examples include:

• sin A cos B

• cos A cos B

• sin A sin B

• 2sin x cos 3x

A sum or difference involves adding or subtracting expressions.

Examples include:

• sin A + sin B

• cos A − cos B

• sin 5x + sin 3x

• cos 7x − cos x

Product-to-sum identities transform multiplication into addition or subtraction.

Sum-to-product identities perform the reverse operation.

Why Are These Transformations Useful?

A trigonometric expression may be difficult to work with in one form but much easier in another.

For example:

sin 5x + sin 3x

is a sum of two trigonometric functions.

Using a sum-to-product identity:

sin 5x + sin 3x

= 2sin 4x cos x

The product form is especially useful when solving equations because a product equals zero whenever at least one of its factors equals zero.

Likewise, converting products to sums is useful in calculus and wave analysis.

Reviewing Compound-Angle Identities

Product-to-sum identities are derived from the sine and cosine addition and subtraction identities.

Recall:

sin(A + B) = sin A cos B + cos A sin B

sin(A − B) = sin A cos B − cos A sin B

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

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

These four relationships are the foundation for the product-to-sum identities.

By adding or subtracting suitable pairs of equations, we can isolate products of trigonometric functions.

Deriving the Product-to-Sum Identity for Sine and Cosine

We want to rewrite:

sin A cos B

as a sum.

Start with the sine addition identity:

sin(A + B) = sin A cos B + cos A sin B

Now write the sine difference identity:

sin(A − B) = sin A cos B − cos A sin B

Add the two equations:

sin(A + B) + sin(A − B)

= sin A cos B + cos A sin B + sin A cos B − cos A sin B

The terms involving cos A sin B cancel.

Therefore:

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

Divide by 2:

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

This is the first product-to-sum identity.

Worked Example

Convert:

sin 5x cos 2x

into a sum.

Use:

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

Let:

A = 5x

B = 2x

Substitute:

sin 5x cos 2x

= [sin(5x + 2x) + sin(5x − 2x)]/2

Simplify:

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

Deriving the Product-to-Sum Identity for Cosine and Sine

Now consider:

cos A sin B

Start with:

sin(A + B) = sin A cos B + cos A sin B

sin(A − B) = sin A cos B − cos A sin B

Subtract the second equation from the first:

sin(A + B) − sin(A − B)

= 2cos A sin B

Divide by 2:

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

This is the second product-to-sum identity.

Worked Example

Convert:

cos 4x sin x

into a sum or difference.

Use:

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

Let:

A = 4x

B = x

Then:

cos 4x sin x

= [sin(4x + x) − sin(4x − x)]/2

Therefore:

cos 4x sin x = (sin 5x − sin 3x)/2

Important Observation

The order of the sine and cosine factors matters when applying the formulas.

Although multiplication is commutative, the angle assigned to A or B must match the chosen identity.

Deriving the Product-to-Sum Identity for Cosine and Cosine

Consider:

cos A cos B

Start with the cosine addition and difference identities:

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

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

Add the equations:

cos(A + B) + cos(A − B)

= 2cos A cos B

Divide by 2:

cos A cos B = [cos(A + B) + cos(A − B)]/2

Worked Example

Convert:

cos 6x cos 2x

into a sum.

Use:

cos A cos B = [cos(A + B) + cos(A − B)]/2

Substitute:

cos 6x cos 2x

= [cos(6x + 2x) + cos(6x − 2x)]/2

Simplify:

= (cos 8x + cos 4x)/2

Deriving the Product-to-Sum Identity for Sine and Sine

Consider:

sin A sin B

Start with:

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

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

Subtract the second equation from the first:

cos(A − B) − cos(A + B)

= 2sin A sin B

Divide by 2:

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

Worked Example

Convert:

sin 5x sin 2x

into a difference.

Use:

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

Substitute:

sin 5x sin 2x

= [cos(5x − 2x) − cos(5x + 2x)]/2

Simplify:

= (cos 3x − cos 7x)/2

Notice that the cosine of the difference comes first.

Product-to-Sum Identity Summary

Recognising Patterns

• Sine × cosine produces a sum or difference of sine functions.

• Cosine × cosine produces a sum of cosine functions.

• Sine × sine produces a difference of cosine functions.

• Every product-to-sum formula contains a factor of 1/2.

These patterns help reduce memorisation.

Converting Products to Sums with Numerical Angles

Product-to-sum identities can also be used to evaluate expressions containing specific angles.

Worked Example

Simplify:

2sin 75° cos 15°

Use:

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

Substitute:

2sin 75° cos 15°

= sin(75° + 15°) + sin(75° − 15°)

= sin 90° + sin 60°

Substitute exact values:

= 1 + √3/2

Therefore:

2sin 75° cos 15° = (2 + √3)/2

Worked Example

Simplify:

2cos 75° cos 15°

Use:

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

Then:

2cos 75° cos 15°

= cos 90° + cos 60°

= 0 + 1/2

Therefore:

2cos 75° cos 15° = 1/2

Worked Example

Simplify:

2sin 75° sin 15°

Use:

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

Then:

= cos 60° − cos 90°

= 1/2 − 0

Therefore:

2sin 75° sin 15° = 1/2

These examples demonstrate how products involving unfamiliar angles can be evaluated using familiar special angles.

Deriving Sum-to-Product Identities

Sum-to-product identities reverse the product-to-sum process.

They allow us to rewrite expressions such as:

sin A + sin B

as products.

The key idea is to introduce two new angles:

u = (A + B)/2

v = (A − B)/2

Then:

A = u + v

B = u − v

This substitution transforms a sum of two angles into a sum and difference involving u and v.

It provides a systematic method for deriving the sum-to-product identities.

Deriving the Sum-to-Product Identity for Sine Addition

Start with:

sin A + sin B

Let:

A = u + v

B = u − v

Then:

sin A + sin B

= sin(u + v) + sin(u − v)

Expand both terms:

= sin u cos v + cos u sin v + sin u cos v − cos u sin v

Combine like terms:

= 2sin u cos v

Substitute:

u = (A + B)/2

v = (A − B)/2

Therefore:

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

Worked Example

Convert:

sin 7x + sin 3x

into a product.

Use:

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

Let:

A = 7x

B = 3x

Substitute:

= 2sin[(7x + 3x)/2]cos[(7x − 3x)/2]

Simplify:

= 2sin(5x)cos(2x)

Therefore:

sin 7x + sin 3x = 2sin 5x cos 2x

Deriving the Sum-to-Product Identity for Sine Subtraction

Start with:

sin A − sin B

Using:

A = u + v

B = u − v

we obtain:

sin A − sin B

= sin(u + v) − sin(u − v)

Expand:

= sin u cos v + cos u sin v − sin u cos v + cos u sin v

Combine:

= 2cos u sin v

Substitute:

u = (A + B)/2

v = (A − B)/2

Therefore:

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

Worked Example

Convert:

sin 9x − sin 3x

into a product.

Use:

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

Substitute:

= 2cos[(9x + 3x)/2]sin[(9x − 3x)/2]

Simplify:

= 2cos 6x sin 3x

Deriving the Sum-to-Product Identity for Cosine Addition

Start with:

cos A + cos B

Substitute:

A = u + v

B = u − v

Then:

cos A + cos B

= cos(u + v) + cos(u − v)

Expand:

= cos u cos v − sin u sin v + cos u cos v + sin u sin v

The sine products cancel.

Therefore:

= 2cos u cos v

Substitute the half-sum and half-difference:

cos A + cos B = 2cos[(A + B)/2]cos[(A − B)/2]

Worked Example

Convert:

cos 8x + cos 2x

into a product.

Use:

cos A + cos B = 2cos[(A + B)/2]cos[(A − B)/2]

Substitute:

= 2cos[(8x + 2x)/2]cos[(8x − 2x)/2]

Simplify:

= 2cos 5x cos 3x

Deriving the Sum-to-Product Identity for Cosine Subtraction

Start with:

cos A − cos B

Substitute:

A = u + v

B = u − v

Then:

cos A − cos B

= cos(u + v) − cos(u − v)

Expand:

= cos u cos v − sin u sin v − cos u cos v − sin u sin v

Combine:

= −2sin u sin v

Substitute:

u = (A + B)/2

v = (A − B)/2

Therefore:

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

Worked Example

Convert:

cos 7x − cos 3x

into a product.

Use:

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

Substitute:

= −2sin[(7x + 3x)/2]sin[(7x − 3x)/2]

Simplify:

= −2sin 5x sin 2x

The negative sign is important.

Sum-to-Product Identity Summary

Recognising Patterns

All four identities use:

Half-sum = (A + B)/2

Half-difference = (A − B)/2

Every sum-to-product identity contains a factor of 2.

The cosine subtraction identity contains a negative sign when written in the form shown above.

Understanding the Half-Sum and Half-Difference

The half-sum and half-difference appear in every sum-to-product identity.

For example:

sin 80° + sin 20°

The half-sum is:

(80° + 20°)/2 = 50°

The half-difference is:

(80° − 20°)/2 = 30°

Therefore:

sin 80° + sin 20°

= 2sin 50° cos 30°

Why Do We Use Half-Angles?

The original angles can be reconstructed as:

A = [(A + B)/2] + [(A − B)/2]

B = [(A + B)/2] − [(A − B)/2]

For A = 80° and B = 20°:

80° = 50° + 30°

20° = 50° − 30°

This structure allows the compound-angle identities to be applied in reverse.

Simplifying Trigonometric Expressions Efficiently

Choosing the correct form of an expression can make simplification much easier.

Worked Example

Simplify:

sin 5x + sin x

Use the sine addition sum-to-product identity:

= 2sin[(5x + x)/2]cos[(5x − x)/2]

Simplify:

= 2sin 3x cos 2x

Worked Example

Simplify:

cos 5x − cos x

Use the cosine subtraction identity:

= −2sin[(5x + x)/2]sin[(5x − x)/2]

Therefore:

= −2sin 3x sin 2x

Worked Example

Simplify:

2sin 4x cos x + 2sin 2x cos x

Convert each product to a sum.

First:

2sin 4x cos x = sin 5x + sin 3x

Second:

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

Therefore:

= sin 5x + 2sin 3x + sin x

Alternatively, factor the original expression first:

2cos x(sin 4x + sin 2x)

Use sum-to-product:

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

Therefore:

= 2cos x(2sin 3x cos x)

= 4sin 3x cos² x

Both results are equivalent. The more useful form depends on the problem.

Recognising Useful Identity Substitutions

Different expression patterns suggest different identities.

When Should We Convert a Product to a Sum?

Product-to-sum is useful when:

• Integrating products of trigonometric functions.

• Comparing waves with different frequencies.

• Simplifying products involving unfamiliar angles.

• Rewriting expressions as combinations of simpler sinusoidal functions.

When Should We Convert a Sum to a Product?

Sum-to-product is useful when:

• Solving equations equal to zero.

• Factoring trigonometric expressions.

• Investigating interference and beats.

• Finding the zeros of sums of sine or cosine functions.

Solving Equations Using Sum-to-Product Identities

One of the most useful applications is solving equations.

Consider:

sin 5x + sin 3x = 0

for:

0° ≤ x < 360°

Convert the Sum to a Product

Use:

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

Then:

sin 5x + sin 3x

= 2sin 4x cos x

Therefore:

2sin 4x cos x = 0

Apply the Zero-Product Property

A product equals zero when at least one factor equals zero.

Therefore:

sin 4x = 0

or:

cos x = 0

Solve the First Equation

sin 4x = 0

Since:

0° ≤ 4x < 1440°

the solutions are:

4x = 0°, 180°, 360°, 540°, 720°, 900°, 1080°, 1260°

Divide by 4:

x = 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°

Solve the Second Equation

cos x = 0

Therefore:

x = 90°, 270°

These solutions already appear in the first set.

Final Answer

x = 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°

The sum-to-product identity transforms a difficult-looking sum into a factorable equation.

Solving Equations Involving Cosine Sums

Worked Example

Solve:

cos 3x + cos x = 0

for:

0° ≤ x < 360°

Convert to a Product

cos 3x + cos x

= 2cos[(3x + x)/2]cos[(3x − x)/2]

Simplify:

= 2cos 2x cos x

Therefore:

2cos 2x cos x = 0

So:

cos 2x = 0

or:

cos x = 0

First Case

cos 2x = 0

Within:

0° ≤ 2x < 720°

the solutions are:

2x = 90°, 270°, 450°, 630°

Divide by 2:

x = 45°, 135°, 225°, 315°

Second Case

cos x = 0

x = 90°, 270°

Final Answer

x = 45°, 90°, 135°, 225°, 270°, 315°

Solving Equations Involving Cosine Differences

Worked Example

Solve:

cos 5x − cos x = 0

for:

0° ≤ x < 360°

Use:

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

Therefore:

cos 5x − cos x

= −2sin 3x sin 2x

Set equal to zero:

−2sin 3x sin 2x = 0

Thus:

sin 3x = 0

or:

sin 2x = 0

First Case

sin 3x = 0

3x = 180°n

x = 60°n

Within the interval:

x = 0°, 60°, 120°, 180°, 240°, 300°

Second Case

sin 2x = 0

2x = 180°n

x = 90°n

Within the interval:

x = 0°, 90°, 180°, 270°

Combine the Solutions

Remove repeated values.

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

Solving Equations Involving Sine Differences

Worked Example

Solve:

sin 4x − sin 2x = 0

for:

0° ≤ x < 360°

Use:

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

Then:

sin 4x − sin 2x

= 2cos 3x sin x

Therefore:

2cos 3x sin x = 0

So:

cos 3x = 0

or:

sin x = 0

First Case

cos 3x = 0

3x = 90°, 270°, 450°, 630°, 810°, 990°

Divide by 3:

x = 30°, 90°, 150°, 210°, 270°, 330°

Second Case

sin x = 0

x = 0°, 180°

Final Answer

x = 0°, 30°, 90°, 150°, 180°, 210°, 270°, 330°

Verifying Product-to-Sum Identities

Identity proofs demonstrate why two expressions are equivalent.

Worked Example

Verify:

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

Start with the right-hand side:

sin(A + B) + sin(A − B)

Expand:

= sin A cos B + cos A sin B + sin A cos B − cos A sin B

Combine like terms:

= 2sin A cos B

Therefore:

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

The identity is verified.

Worked Example

Verify:

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

Start with:

cos(A + B) + cos(A − B)

Expand:

= cos A cos B − sin A sin B + cos A cos B + sin A sin B

Simplify:

= 2cos A cos B

The identity is verified.

Verifying Sum-to-Product Identities

Worked Example

Verify:

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

Let:

u = (A + B)/2

v = (A − B)/2

Then:

A = u + v

B = u − v

Start with the left-hand side:

sin A + sin B

= sin(u + v) + sin(u − v)

Expand:

= sin u cos v + cos u sin v + sin u cos v − cos u sin v

Combine:

= 2sin u cos v

Substitute the original expressions:

= 2sin[(A + B)/2]cos[(A − B)/2]

Therefore, the identity is verified.

Why This Proof Matters

It demonstrates how a substitution can simplify the structure of a complicated expression.

The proof also shows that sum-to-product identities follow directly from compound-angle identities.

Product-to-Sum Identities and Double Angles

When both angles are equal, product-to-sum identities produce familiar double-angle and power-reduction identities.

Sine and Cosine

Start with:

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

Let:

A = B = x

Then:

sin x cos x

= [sin 2x + sin 0]/2

Since sin 0 = 0:

sin x cos x = sin 2x/2

Therefore:

2sin x cos x = sin 2x

Cosine Squared

Start with:

cos A cos B = [cos(A + B) + cos(A − B)]/2

Let:

A = B = x

Then:

cos² x = [cos 2x + cos 0]/2

Since cos 0 = 1:

cos² x = (1 + cos 2x)/2

Sine Squared

Start with:

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

Let:

A = B = x

Then:

sin² x = [cos 0 − cos 2x]/2

Therefore:

sin² x = (1 − cos 2x)/2

These derivations show how product-to-sum identities connect with other important trigonometric relationships.

Applications in Physics: Wave Interference

Product-to-sum and sum-to-product identities are useful for analysing the combination of waves.

Suppose two waves have the same amplitude A but slightly different angular frequencies:

y₁(t) = A sin(ω₁t)

y₂(t) = A sin(ω₂t)

The combined displacement is:

y(t) = A sin(ω₁t) + A sin(ω₂t)

Factor A:

y(t) = A[sin(ω₁t) + sin(ω₂t)]

Use the sine sum-to-product identity:

y(t) = 2A sin[(ω₁ + ω₂)t/2]cos[(ω₁ − ω₂)t/2]

This expression describes a rapidly oscillating wave whose amplitude varies more slowly.

The phenomenon is called beats.

Understanding Beats

The term:

sin[(ω₁ + ω₂)t/2]

describes the rapid oscillation.

The term:

cos[(ω₁ − ω₂)t/2]

controls the slowly varying amplitude envelope.

For two frequencies f₁ and f₂, the beat frequency is:

f_beat = |f₁ − f₂|

Example

Two tuning forks produce frequencies of 440 Hz and 444 Hz.

The beat frequency is:

f_beat = |444 − 440|

f_beat = 4 Hz

The combined sound becomes louder and quieter four times per second.

Applications in Signal Processing

Products of sinusoidal signals appear in communication systems.

Suppose two signals are multiplied:

y(t) = cos(ω₁t)cos(ω₂t)

Use the product-to-sum identity:

cos A cos B = [cos(A + B) + cos(A − B)]/2

Therefore:

y(t) = [cos((ω₁ + ω₂)t) + cos((ω₁ − ω₂)t)]/2

The product contains two frequency components:

• A sum-frequency component.

• A difference-frequency component.

This principle is important in:

• Radio communication.

• Frequency mixing.

• Signal modulation.

• Electronic circuits.

• Audio processing.

Applications in Calculus: Integrating Products

Product-to-sum identities can make certain trigonometric integrals much easier.

Worked Example

Evaluate:

∫ sin 3x cos 2x dx

Use:

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

Therefore:

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

Substitute into the integral:

∫ sin 3x cos 2x dx

= (1/2)∫(sin 5x + sin x) dx

Integrate each term:

= (1/2)[−cos 5x/5 − cos x] + C

Simplify:

= −cos 5x/10 − cos x/2 + C

This method avoids more complicated integration techniques.

Worked Example

Evaluate:

∫ cos 4x cos 2x dx

Use:

cos 4x cos 2x = [cos 6x + cos 2x]/2

Therefore:

∫ cos 4x cos 2x dx

= (1/2)∫(cos 6x + cos 2x) dx

Integrate:

= (1/2)[sin 6x/6 + sin 2x/2] + C

Therefore:

= sin 6x/12 + sin 2x/4 + C

Applications in Engineering: Alternating Current

In electrical circuits, voltage and current may vary sinusoidally.

Suppose:

v(t) = V₀ cos(ωt)

i(t) = I₀ cos(ωt − φ)

Instantaneous electrical power is:

p(t) = v(t)i(t)

Substitute:

p(t) = V₀I₀ cos(ωt)cos(ωt − φ)

Use the product-to-sum identity:

cos A cos B = [cos(A + B) + cos(A − B)]/2

Let:

A = ωt

B = ωt − φ

Then:

A + B = 2ωt − φ

A − B = φ

Therefore:

p(t) = (V₀I₀/2)[cos(2ωt − φ) + cos φ]

The expression contains:

• A time-varying component.

• A constant component.

Over a complete cycle, the time-varying cosine term has an average of zero.

Therefore, the average power is:

P_avg = (V₀I₀/2)cos φ

This is an important relationship in alternating-current circuit analysis.

Choosing the Most Efficient Method

Sometimes more than one identity can be applied to an expression.

The best choice depends on the goal.

If the Goal Is to Simplify

Look for familiar patterns and choose the shortest transformation.

Example:

2sin x cos x

can be simplified directly to:

sin 2x

There is no need to use the general product-to-sum formula.

If the Goal Is to Solve an Equation

Sum-to-product is often useful because it produces factors.

Example:

sin 5x − sin x = 0

becomes:

2cos 3x sin 2x = 0

The zero-product property can then be applied.

If the Goal Is to Integrate

Product-to-sum is often useful because sums of sine and cosine functions are easier to integrate.

Example:

cos 5x cos 2x

becomes:

(cos 7x + cos 3x)/2

If the Goal Is to Analyse Waves

Sum-to-product can reveal amplitude envelopes and beat patterns.

Product-to-sum can reveal sum and difference frequencies.

Common Mistakes

Confusing product-to-sum with sum-to-product

Product-to-sum starts with multiplication. Sum-to-product starts with addition or subtraction.

Forgetting the factor of 1/2

Every standard product-to-sum identity contains a factor of 1/2.

Forgetting the factor of 2

Every standard sum-to-product identity contains a factor of 2.

Using the wrong sign for sine products

The correct identity is:

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

Forgetting the negative sign in cosine subtraction

The identity is:

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

Confusing the half-sum and half-difference

The half-sum is (A + B)/2.

The half-difference is (A − B)/2.

Failing to simplify angles correctly

For example:

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

not 10x.

Mixing degrees and radians

All angles in a calculation must use consistent units.

Forgetting periodic solutions

Trigonometric equations may have several solutions within a specified interval.

Failing to adjust intervals for multiple angles

When solving sin 4x = 0, the argument 4x covers four times the interval of x.

Repeating solutions

Different factors may produce the same solution. List each solution only once.

Using numerical examples as complete proofs

A complete identity proof requires valid algebraic reasoning.

Key Terms

Product-to-Sum Identity: A trigonometric identity that converts a product of trigonometric functions into a sum or difference.

Sum-to-Product Identity: A trigonometric identity that converts a sum or difference of trigonometric functions into a product.

Compound-Angle Identity: A formula expressing a trigonometric function of a sum or difference of angles.

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Product: The result of multiplying two quantities.

Sum: The result of adding two quantities.

Difference: The result of subtracting one quantity from another.

Half-Sum: Half the sum of two angles, (A + B)/2.

Half-Difference: Half the difference of two angles, (A − B)/2.

Equivalent Expressions: Expressions that have the same value over their common domain.

Identity Substitution: Replacing an expression with an equivalent expression using a known identity.

Simplification: Rewriting an expression in a more useful or less complicated form.

Factorisation: Expressing an algebraic expression as a product of factors.

Zero-Product Property: The principle that a product equals zero when at least one factor equals zero.

Reference Angle: The acute angle between an angle's terminal side and the x-axis.

Periodicity: The property of a function whose values repeat at regular intervals.

Period: The interval required for a periodic function to complete one cycle.

Amplitude: The maximum displacement of a sinusoidal function from its midline.

Frequency: The number of complete cycles per unit time.

Angular Frequency: The rate of angular change, commonly represented by ω.

Wave Interference: The combination of two or more waves to produce a resultant wave.

Constructive Interference: Interference that increases the resultant amplitude.

Destructive Interference: Interference that reduces the resultant amplitude.

Beats: Periodic variations in amplitude produced by combining waves with slightly different frequencies.

Beat Frequency: The absolute difference between the frequencies of two interfering waves.

Amplitude Envelope: A curve describing the variation in the maximum amplitude of a wave.

Frequency Mixing: The process of producing sum and difference frequency components by multiplying signals.

Signal Modulation: The process of varying a signal to carry information.

Trigonometric Integration: The use of integration techniques involving trigonometric functions.

Instantaneous Power: The electrical power at a particular moment, calculated as voltage multiplied by current.

Average Power: The mean power transferred over a specified time interval.

Phase Difference: The angular difference between two periodic signals.

Mathematical Proof: A logical sequence of justified steps demonstrating that a statement is true.

Check Your Understanding

1. Explain the difference between product-to-sum and sum-to-product identities.

2. Derive the product-to-sum identity for sin A cos B using the sine addition and subtraction identities.

3. Derive the product-to-sum identity for cos A cos B.

4. Convert the following expression into a sum:

sin 6x cos 2x

5. Convert the following expression into a sum or difference:

sin 5x sin 3x

6. Convert the following expression into a sum:

cos 7x cos x

7. Derive the sum-to-product identity for sin A + sin B.

8. Convert the following expression into a product:

sin 9x + sin 3x

9. Convert the following expression into a product:

cos 8x − cos 2x

10. Convert the following expression into a product:

sin 7x − sin x

11. Use a product-to-sum identity to determine the exact value of:

2sin 75° cos 15°

12. Solve:

sin 5x + sin 3x = 0

for 0° ≤ x < 360°.

13. Solve:

cos 4x − cos 2x = 0

for 0° ≤ x < 360°.

14. Verify:

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

Show each algebraic step.

15. Two sound waves have frequencies of 500 Hz and 506 Hz. Determine the beat frequency and explain how the sum-to-product identity describes the resulting wave.

Key Takeaways

• Product-to-sum identities convert products of trigonometric functions into sums or differences.

• Sum-to-product identities convert sums or differences into products.

• Both groups of identities are derived from compound-angle identities.

• The product sin A cos B can be rewritten as [sin(A + B) + sin(A − B)]/2.

• The product cos A sin B can be rewritten as [sin(A + B) − sin(A − B)]/2.

• The product cos A cos B can be rewritten as [cos(A + B) + cos(A − B)]/2.

• The product sin A sin B can be rewritten as [cos(A − B) − cos(A + B)]/2.

• Every standard product-to-sum identity contains a factor of 1/2.

• Every standard sum-to-product identity contains a factor of 2.

• Sum-to-product identities use the half-sum and half-difference of two angles.

• The sum sin A + sin B becomes 2sin[(A + B)/2]cos[(A − B)/2].

• The difference sin A − sin B becomes 2cos[(A + B)/2]sin[(A − B)/2].

• The sum cos A + cos B becomes 2cos[(A + B)/2]cos[(A − B)/2].

• The difference cos A − cos B becomes −2sin[(A + B)/2]sin[(A − B)/2].

• Recognising identity patterns allows complicated expressions to be simplified efficiently.

• Sum-to-product identities are particularly useful when solving trigonometric equations equal to zero.

• The zero-product property allows factorised trigonometric equations to be solved systematically.

• Product-to-sum identities are useful for integrating products of trigonometric functions.

• These identities are closely related to double-angle and power-reduction identities.

• Sum-to-product identities help explain beats and interference between waves.

• Product-to-sum identities reveal sum and difference frequencies in signal processing.

• The identities are used in alternating-current circuits, wave analysis, engineering, and calculus.

• Mastering product-to-sum and sum-to-product identities strengthens trigonometric reasoning and provides valuable techniques for advanced mathematics and physics.