Composite Trigonometric Functions

Trigonometric functions are often combined to create more complex expressions and models. Instead of working with only:

sin(x)

or:

cos(x)

we might encounter expressions such as:

sin(x) + cos(x)

2sin(x) − cos(x)

sin²(x) + cos²(x)

sin(2x)cos(x)

or even a true composition of functions, such as:

sin(cos(x))

These combinations are called composite or combined trigonometric expressions in many courses. They allow us to describe more complicated patterns than a single basic sine, cosine, or tangent function.

Two Meanings of "Composite"

The word composite needs some care because it can describe two related but different ideas.

Combining Trigonometric Functions

Functions may be added, subtracted, multiplied, or divided:

sin(x) + cos(x)

sin(x)cos(x)

sin(x)/cos(x)

These are combinations of trigonometric functions.

Function Composition

One function can also be placed inside another:

sin(cos(x))

This is a true composite function.

If:

f(x) = sin(x)

and:

g(x) = cos(x)

then:

f(g(x)) = sin(cos(x))

Both ideas are useful, so we will examine both.

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Evaluating Combined Trigonometric Functions

Consider:

f(x) = sin(x) + cos(x)

To evaluate f(x), calculate both trigonometric values and then add them.

Suppose:

x = 0

Then:

f(0) = sin(0) + cos(0)

= 0 + 1

= 1

At:

x = π/2

f(π/2) = sin(π/2) + cos(π/2)

= 1 + 0

= 1

At:

x = π

f(π) = sin(π) + cos(π)

= 0 − 1

= −1

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Important Exact Trigonometric Values

Composite expressions frequently use familiar angles, so exact values are valuable.

Knowing these values allows many expressions to be evaluated without a calculator.

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Worked Example: Addition

Evaluate:

2sin(x) + cos(x)

when:

x = π/6

Use:

sin(π/6) = 1/2

cos(π/6) = √3/2

Therefore:

2(1/2) + √3/2

= 1 + √3/2

So the exact answer is:

1 + √3/2

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Worked Example: Subtraction

Evaluate:

3cos(x) − 2sin(x)

when:

x = π/3

Use:

cos(π/3) = 1/2

sin(π/3) = √3/2

Substitute:

3(1/2) − 2(√3/2)

= 3/2 − √3

Therefore:

3cos(π/3) − 2sin(π/3) = 3/2 − √3

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Multiplying Trigonometric Functions

Functions can also be multiplied.

Consider:

f(x) = sin(x)cos(x)

At:

x = π/4

sin(π/4) = √2/2

cos(π/4) = √2/2

Therefore:

f(π/4) = (√2/2)(√2/2)

= 2/4

= 1/2

Products of trigonometric functions appear frequently in identities, calculus, waves, and physics.

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Dividing Trigonometric Functions

Consider:

sin(x)/cos(x)

One of the fundamental trigonometric identities tells us:

sin(x)/cos(x) = tan(x)

as long as:

cos(x) ≠ 0

Therefore:

sin(x)/cos(x)

can often be simplified immediately to:

tan(x)

This also reminds us that the expression is undefined wherever cos(x) = 0.

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Fundamental Trigonometric Identities

Identities are extremely useful when simplifying composite expressions.

Three of the most important are:

sin²(x) + cos²(x) = 1

1 + tan²(x) = sec²(x)

1 + cot²(x) = csc²(x)

These are called Pythagorean identities.

They are true for every value of x for which the expressions are defined.

Simplifying with an Identity

Simplify:

sin²(x) + cos²(x) + sin(x)

Use:

sin²(x) + cos²(x) = 1

Therefore:

sin²(x) + cos²(x) + sin(x)

= 1 + sin(x)

A complicated-looking expression has become much simpler.

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Another Simplification

Simplify:

1 − sin²(x)

From:

sin²(x) + cos²(x) = 1

subtract sin²(x):

cos²(x) = 1 − sin²(x)

Therefore:

1 − sin²(x) = cos²(x)

Similarly:

1 − cos²(x) = sin²(x)

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Quotient Identities

Two important quotient identities are:

tan(x) = sin(x)/cos(x)

and:

cot(x) = cos(x)/sin(x)

These allow expressions containing fractions to be simplified.

For example:

cos(x)tan(x)

Replace tan(x):

cos(x) × sin(x)/cos(x)

Cancel cos(x), where cos(x) ≠ 0:

= sin(x)

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Reciprocal Identities

The reciprocal trigonometric functions are:

sec(x) = 1/cos(x)

csc(x) = 1/sin(x)

cot(x) = 1/tan(x)

These relationships are useful when expressions contain several different trigonometric functions.

For example:

cos(x)sec(x)

= cos(x) × 1/cos(x)

= 1

where cos(x) ≠ 0.

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Evaluating True Composite Functions

Now consider actual function composition.

Suppose:

f(x) = sin(x)

and:

g(x) = x²

Then:

f(g(x)) = sin(x²)

The output of g becomes the input of f.

To evaluate at x = 2:

g(2) = 2²

= 4

Then:

f(4) = sin(4)

Therefore:

f(g(2)) = sin(4)

If working in radians, a calculator gives approximately:

−0.757

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The Inside Function Comes First

Consider:

sin(cos(x))

To evaluate this expression:

1. calculate cos(x)
2. use that answer as the input of sine

Suppose:

x = 0

First:

cos(0) = 1

Then:

sin(1) ≈ 0.841

Therefore:

sin(cos(0)) ≈ 0.841

Notice that the 1 is interpreted as 1 radian.

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Order Matters in Function Composition

In general:

f(g(x)) ≠ g(f(x))

For example:

sin(cos(x))

is not generally equal to:

cos(sin(x))

At x = 0:

sin(cos(0)) = sin(1)

≈ 0.841

but:

cos(sin(0)) = cos(0)

= 1

Therefore, the order of composition matters.

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Composition Notation

Function composition is often written using the symbol:

∘

If:

f(x) = sin(x)

and:

g(x) = 2x

then:

(f ∘ g)(x) = f(g(x))

Therefore:

(f ∘ g)(x) = sin(2x)

But:

(g ∘ f)(x) = g(f(x))

so:

(g ∘ f)(x) = 2sin(x)

These are different functions.

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Combined Functions and Graphs

When two trigonometric functions are combined, their graphs can interact in interesting ways.

Consider:

y = sin(x) + cos(x)

At every x-value, the y-values of the two functions are added.

For example:

The resulting function is still periodic.

A Surprising Result

The expression:

sin(x) + cos(x)

can actually be rewritten as a single sinusoidal function:

√2 sin(x + π/4)

This means that adding two sinusoidal functions with the same frequency can sometimes produce another sinusoidal function.

Its amplitude is:

√2

rather than 1.

This idea becomes extremely important when studying waves.

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Combining a sin(x) + b cos(x)

A more general expression is:

a sin(x) + b cos(x)

This can be rewritten in the form:

R sin(x + φ)

where:

R = √(a² + b²)

The value R becomes the amplitude of the resulting sinusoidal function.

For example:

3sin(x) + 4cos(x)

has:

R = √(3² + 4²)

= √25

= 5

So the combined function has amplitude:

5

The phase angle φ depends on the relative contributions of the sine and cosine terms.

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Rewriting a Combination as One Function

Suppose:

3sin(x) + 4cos(x) = R sin(x + φ)

Expand the right side using:

sin(x + φ) = sin(x)cos(φ) + cos(x)sin(φ)

Then:

R sin(x + φ)

= Rcos(φ)sin(x) + Rsin(φ)cos(x)

Compare coefficients:

Rcos(φ) = 3

Rsin(φ) = 4

Since R = 5:

cos(φ) = 3/5

sin(φ) = 4/5

Therefore:

φ ≈ 0.927 radians

or:

φ ≈ 53.1°

So:

3sin(x) + 4cos(x) = 5sin(x + 53.1°)

when working in degrees.

This technique is sometimes called harmonic addition or the auxiliary angle method.

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Why This Simplification Is Useful

Suppose we have:

3sin(x) + 4cos(x)

It may not be immediately obvious what its maximum value is.

But once rewritten as:

5sin(x + φ)

we immediately know:

maximum = 5

minimum = −5

amplitude = 5

This makes graphing and solving equations much easier.

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Different Frequencies

Not every combination can be simplified into one ordinary sine wave.

Consider:

sin(x) + sin(2x)

The two terms have different frequencies.

The first has period:

2π

The second has period:

π

Their sum produces a more complicated periodic shape.

Finding the Period of a Combined Function

Consider:

f(x) = sin(x) + sin(2x)

sin(x) repeats every:

2π

sin(2x) repeats every:

π

We need a length over which both functions repeat.

The smallest such interval is:

2π

Therefore, the combined function has period:

2π

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Another Period Example

Consider:

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

Periods:

sin(2x):

π

cos(4x):

π/2

Both functions return to their starting values after π.

Therefore, the combined function has period:

π

This is similar to finding a common multiple.

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Graphical Interpretation

To understand:

y = f(x) + g(x)

imagine adding the vertical values of the two graphs at every x-coordinate.

If:

f(x) = 2

and:

g(x) = 3

then:

f(x) + g(x) = 5

If:

f(x) = 2

and:

g(x) = −3

then:

f(x) + g(x) = −1

The resulting graph depends on how the two component functions interact.

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Constructive and Destructive Combination

When two waves are combined, their values may reinforce or oppose each other.

If both are positive at the same time, their sum becomes larger.

If both are negative, their combined displacement becomes more negative.

If one is positive and the other negative, they may partly or completely cancel.

This idea is closely connected to wave superposition.

Composite Trigonometric Functions and Waves

One of the most important applications of combined trigonometric functions is wave modelling.

Real waves are often not perfect sine waves.

A complex wave can sometimes be represented as a combination such as:

y = sin(x) + 0.5sin(2x) + 0.25sin(3x)

Each term contributes a different frequency and amplitude.

Together, they produce a more complex waveform.

This idea is fundamental to:

• sound
• music
• acoustics
• communications
• electronics
• signal processing
• optics
• vibration analysis

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Beats

An interesting pattern occurs when two waves have frequencies that are close but not identical.

For example:

sin(10x) + sin(11x)

The waves repeatedly move into and out of phase.

This creates alternating regions of stronger and weaker amplitude called beats.

Beats can be heard when two musical notes have slightly different frequencies.

Product-to-Sum Identity

Products of trigonometric functions can sometimes be rewritten as sums.

For example:

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

If A = B = x:

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

Therefore:

sin(x)cos(x) = (1/2)sin(2x)

This identity can make products easier to analyze.

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Double-Angle Identities

Some common composite expressions can be simplified using double-angle identities.

Sine

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

Cosine

cos(2x) = cos²(x) − sin²(x)

Other equivalent forms are:

cos(2x) = 2cos²(x) − 1

and:

cos(2x) = 1 − 2sin²(x)

These identities allow us to move between different forms of the same expression.

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Example: Simplifying with a Double-Angle Identity

Simplify:

4sin(x)cos(x)

Use:

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

Then:

4sin(x)cos(x)

= 2[2sin(x)cos(x)]

= 2sin(2x)

This form may be easier to graph or solve.

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Example: Simplifying a Squared Expression

Simplify:

2cos²(x) − 1

Using the double-angle identity:

cos(2x) = 2cos²(x) − 1

Therefore:

2cos²(x) − 1 = cos(2x)

A quadratic-looking trigonometric expression becomes a single cosine function.

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Factoring Trigonometric Expressions

Ordinary algebraic techniques also apply.

Consider:

sin²(x) + sin(x)cos(x)

Both terms contain sin(x).

Factor:

sin(x)[sin(x) + cos(x)]

This may be useful when solving equations.

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Solving a Combined Trigonometric Equation

Solve:

sin²(x) + sin(x)cos(x) = 0

Factor:

sin(x)[sin(x) + cos(x)] = 0

Therefore either:

sin(x) = 0

or:

sin(x) + cos(x) = 0

The second equation gives:

sin(x) = −cos(x)

Divide by cos(x), where appropriate:

tan(x) = −1

We can then solve each equation over the required domain.

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Solving sin(x) + cos(x) = 0

Suppose:

0 ≤ x < 2π

We have:

sin(x) = −cos(x)

Therefore:

tan(x) = −1

Tangent is negative in Quadrants II and IV.

So:

x = 3π/4

and:

x = 7π/4

These are the solutions on the specified interval.

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Solving Using a Combined Form

Suppose:

sin(x) + cos(x) = 1

Rewrite:

√2 sin(x + π/4) = 1

Divide:

sin(x + π/4) = 1/√2

= √2/2

This converts the original equation into a familiar trigonometric equation.

The combined form can therefore make equations much easier to solve.

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Domain Restrictions

When simplifying composite trigonometric expressions, always check whether a denominator could equal zero.

For example:

sin(x)/cos(x)

can be simplified to:

tan(x)

but the original expression requires:

cos(x) ≠ 0

Therefore:

x ≠ π/2 + kπ

for integer k.

Simplification should not cause us to forget restrictions in the original expression.

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Composite Functions and Domain

True function composition also requires us to think carefully about domain.

Consider:

arcsin(2x)

The input to arcsin must satisfy:

−1 ≤ 2x ≤ 1

Therefore:

−1/2 ≤ x ≤ 1/2

The domain of a composite function depends on whether the output of the inside function is allowed as an input to the outside function.

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Example with an Inverse Function

Consider:

sin(arcsin(x))

For:

−1 ≤ x ≤ 1

arcsin(x) returns an angle whose sine is x.

Therefore:

sin(arcsin(x)) = x

For example:

sin(arcsin(0.6))

= 0.6

This is an example of a function and its inverse undoing each other.

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But Order Still Matters

Now consider:

arcsin(sin(x))

It is tempting to say this always equals x.

It does not.

The inverse sine function returns values only within its principal range:

−π/2 ≤ arcsin(x) ≤ π/2

Therefore:

arcsin(sin(x)) = x

only for appropriate x-values within that range.

This is an important distinction in advanced trigonometry.

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Graphing Composite Functions

When interpreting a composite or combined graph, ask:

1. What are the component functions?
2. What are their amplitudes?
3. What are their periods?
4. Are their frequencies the same?
5. Are they in phase or shifted?
6. Are they being added, subtracted, multiplied, or composed?
7. Where do they reinforce one another?
8. Where do they cancel?
9. What is the period of the resulting function?
10. Can the expression be simplified using an identity?

These questions help reveal the structure behind a complicated graph.

Real-World Example: Sound

Suppose a simplified sound signal is:

S(t) = 3sin(200πt) + sin(400πt)

The signal contains two oscillations.

The first term has a larger amplitude.

The second term has a higher frequency.

When combined, the resulting waveform is more complex than either component alone.

This helps explain why musical instruments can produce complicated sounds even when they play a single perceived note.

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Real-World Example: Ocean Waves

A water surface may experience several waves simultaneously.

A simplified model might be:

h(t) = 1.2sin(t) + 0.5sin(3t)

The first wave has the larger amplitude.

The second oscillates more rapidly.

At some times the waves reinforce each other.

At other times they partially cancel.

The observed water level is the combined result.

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Real-World Example: Alternating Signals

Electrical signals can contain multiple sinusoidal components.

A simplified voltage model might be:

V(t) = 10sin(ωt) + 2sin(3ωt)

The first term is called the fundamental component.

The second has three times its frequency and is an example of a harmonic.

Analyzing these components helps engineers understand complex electrical signals.

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Connections to Fourier Analysis

One of the most powerful ideas in advanced mathematics is that many complicated periodic functions can be represented approximately by adding sine and cosine functions.

A representation might look like:

f(x) = a₀ + a₁cos(x) + b₁sin(x) + a₂cos(2x) + b₂sin(2x) + ...

This idea leads to Fourier series.

Fourier methods are fundamental in:

• audio processing
• image processing
• telecommunications
• medical imaging
• vibration analysis
• astronomy
• engineering
• physics

A complicated signal can often be understood by examining the simpler sinusoidal components that make it up.

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A Strategy for Composite Problems

When faced with a complicated trigonometric expression, do not immediately start calculating.

Instead:

Identify → Simplify → Evaluate → Interpret

Identify

Determine which trigonometric functions are present.

Simplify

Look for:

• common factors
• Pythagorean identities
• quotient identities
• reciprocal identities
• double-angle identities
• sum or difference identities

Evaluate

Substitute values only after simplifying when possible.

Interpret

If the problem involves a graph or real-world situation, explain what the result means.

This approach is usually more efficient and less error-prone.

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Common Mistakes

Treating addition like multiplication

sin(x) + cos(x)

cannot simply become:

sin(2x)

Assuming sin(x)cos(x) = sin(x²)

These are completely different expressions.

Forgetting order in composition

sin(cos(x))

is generally not equal to:

cos(sin(x))

Evaluating the outside function first

For:

f(g(x))

evaluate g(x) first.

Forgetting identities

Expressions such as:

sin²(x) + cos²(x)

should immediately suggest:

1

Cancelling terms incorrectly

You may cancel common factors, not terms joined by addition or subtraction.

Ignoring domain restrictions

Expressions containing denominators may be undefined at certain values.

Mixing degrees and radians

Check the expected angle mode before evaluating numerically.

Assuming every sum of waves is a simple sine wave

Functions with different frequencies often produce more complex shapes.

Confusing amplitude with maximum value

A combined function may have a vertical shift, so amplitude and maximum value are not necessarily the same.

Assuming all composite functions are periodic

Composition does not automatically guarantee periodicity.

Ignoring graphical meaning

An expression may be easier to understand by examining how its component functions interact.

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Key Terms

Composite function — A function formed by using the output of one function as the input of another.

Function composition — The process of applying one function to the result of another.

Combined function — A function formed by adding, subtracting, multiplying, or dividing functions.

Component function — One of the individual functions making up a combined expression.

Trigonometric identity — An equation involving trigonometric functions that is true wherever both sides are defined.

Pythagorean identity — An identity based on the relationship sin²(x) + cos²(x) = 1.

Quotient identity — An identity expressing tangent or cotangent as a quotient of sine and cosine.

Reciprocal identity — An identity connecting functions such as cosine and secant or sine and cosecant.

Double-angle identity — An identity involving an angle and twice that angle.

Product-to-sum identity — An identity that rewrites products of trigonometric functions as sums.

Exact value — A value expressed precisely rather than as a decimal approximation.

Period — The interval over which a periodic function repeats.

Frequency — The number of cycles completed per unit interval.

Amplitude — The maximum displacement from the midline of a sinusoidal function.

Phase — The position of a periodic function within its cycle.

Superposition — The combination of two or more waves by adding their displacements.

Constructive interference — Reinforcement that occurs when wave displacements combine in the same direction.

Destructive interference — Reduction that occurs when wave displacements oppose one another.

Beat — A repeating variation in amplitude caused by combining waves with similar frequencies.

Harmonic — A frequency that is an integer multiple of a fundamental frequency.

Fundamental frequency — The lowest main frequency in a periodic signal.

Auxiliary angle method — A method for rewriting a sin(x) + b cos(x) as one sinusoidal function.

Domain restriction — A value excluded because an expression or function is undefined there.

Inverse trigonometric function — A function such as arcsin, arccos, or arctan used to determine angles from trigonometric ratios.

Fourier series — A representation of a periodic function using a sum of sine and cosine terms.

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Key Takeaways

• Trigonometric functions can be added, subtracted, multiplied, divided, or composed.
• A true composite function places one function inside another.
• In f(g(x)), evaluate the inside function first.
• Function composition is generally not commutative:

f(g(x)) ≠ g(f(x))

• Exact trigonometric values can simplify calculations.
• Identities allow complicated expressions to be rewritten in simpler forms.
• One of the most important identities is:

sin²(x) + cos²(x) = 1

• Tangent can be written as:

tan(x) = sin(x)/cos(x)

• Products such as sin(x)cos(x) can be simplified using double-angle identities.
• Algebraic techniques such as factoring still apply to trigonometric expressions.
• Expressions of the form:

a sin(x) + b cos(x)

can be rewritten as a single sinusoidal function.

• The resulting amplitude is:

√(a² + b²)

• Functions with different frequencies can create complex periodic patterns.
• The period of a combined function must allow all component functions to repeat.
• Combined waves can reinforce or cancel one another.
• This behaviour is related to constructive and destructive interference.
• Waves with similar frequencies can produce beats.
• Composite trigonometric models appear in sound, water waves, electronics, vibrations, and signal processing.
• Domain restrictions must be preserved when simplifying expressions.
• Inverse trigonometric compositions require particular care with domains and ranges.
• Complicated periodic signals can often be understood as combinations of simpler sine and cosine waves.
• This principle leads to Fourier analysis, one of the most important applications of trigonometric functions in science and engineering.

A useful strategy is:

Identify → Simplify → Evaluate → Interpret

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Check Your Understanding

1. Evaluate:

sin(0) + cos(0)

2. Evaluate:

sin(π/2) + cos(π/2)

3. Evaluate:

2sin(π/6) + cos(π/6)

Give an exact answer.

4. Evaluate:

sin(π/4)cos(π/4)

5. Simplify:

sin²(x) + cos²(x)

6. Simplify:

1 − sin²(x)

7. Simplify:

1 − cos²(x)

8. Simplify:

sin(x)/cos(x)

9. Simplify:

cos(x)tan(x)

10. Simplify:

cos(x)sec(x)

11. Simplify:

2sin(x)cos(x)

12. Simplify:

2cos²(x) − 1

13. If:

f(x) = sin(x)

and:

g(x) = 2x

find:

f(g(x))

14. For the same functions, find:

g(f(x))

15. Explain why your answers to Questions 13 and 14 are different.

16. Evaluate:

sin(cos(0))

approximately.

17. Evaluate:

cos(sin(0))

18. Simplify by factoring:

sin²(x) + sin(x)cos(x)

19. Solve:

sin(x) + cos(x) = 0

for:

0 ≤ x < 2π

20. What is the amplitude of:

3sin(x) + 4cos(x)

after it is rewritten as one sinusoidal function?

21. Rewrite:

sin(x) + cos(x)

as one sinusoidal function.

22. Find the period of:

sin(x) + sin(2x)

23. Find the period of:

sin(2x) + cos(4x)

24. Explain why:

sin(x) + sin(2x)

cannot normally be rewritten as a single basic sine wave with one frequency.

25. Explain what constructive interference means.

26. Explain what destructive interference means.

27. What are beats, and why can they occur when two waves are combined?

28. Explain why domain restrictions must still be considered after simplifying a trigonometric expression.

29. Describe one real-world situation in which several trigonometric functions might be combined.

30. Challenge: Consider:

f(x) = 3sin(x) + 4cos(x)

a. Find the amplitude when the function is rewritten as one sinusoidal function.
b. Find the phase angle φ if the function is written as 5sin(x + φ).
c. Write the resulting single sinusoidal function.
d. State its maximum value.
e. State its minimum value.
f. State its period.
g. Explain why combining the original sine and cosine functions does not change the period in this case.
h. Explain how the graph of the combined function differs from the graph of y = sin(x).
i. Give one real-world situation where combining sinusoidal functions would be useful.
j. Explain how this example connects to the idea of wave superposition.