3. Systems of Trigonometric Equations

Systems of Trigonometric Equations

A system of trigonometric equations consists of two or more equations involving trigonometric functions that must be satisfied simultaneously.

Unlike a single trigonometric equation, where we find values that satisfy one relationship, a system requires us to find values that satisfy every equation at the same time.

For example:

sin x + cos y = 1

sin x − cos y = 0

Both equations contain trigonometric expressions involving two unknown angles. To solve the system, we must find every ordered pair (x, y) that makes both equations true.

Systems of trigonometric equations can be solved using familiar algebraic techniques, including substitution and elimination. However, the periodic nature of trigonometric functions means that systems may have several solutions, infinitely many solutions, or no solutions.

These equations are useful in physics, engineering, navigation, wave analysis, and the study of periodic phenomena.

Understanding Simultaneous Trigonometric Equations

A simultaneous trigonometric system may involve one or more unknown angles.

For example:

System A

sin x = cos x

sin 2x = 1

This system contains one unknown angle, x, and requires a value that satisfies both equations.

System B

sin x + cos y = 1

sin x − cos y = 0

This system contains two unknown angles, x and y.

System C

sin x + sin y = 1

sin x − sin y = 0

This system contains two trigonometric expressions that can be isolated using algebraic elimination.

What Counts as a Solution?

For a system involving two variables, a solution is an ordered pair:

(x, y)

The first coordinate represents x, and the second represents y.

For example:

(30°, 60°)

and:

(60°, 30°)

are different ordered pairs.

Even when both pairs contain the same two angles, they may not both satisfy a particular system.

Important Principle

A valid solution must satisfy every equation in the system, not just one.

Reviewing the Algebraic Methods

The techniques used for linear simultaneous equations also apply to many trigonometric systems.

Substitution

Substitution involves isolating one variable or expression in an equation and replacing it in another equation.

For example:

sin x + cos y = 1

sin x = 2cos y

Substitute the second equation into the first:

2cos y + cos y = 1

3cos y = 1

cos y = 1/3

Once y is determined, substitute back to find x.

Elimination

Elimination involves adding or subtracting equations to remove one expression.

For example:

sin x + cos y = 1

sin x − cos y = 0

Add the equations:

2sin x = 1

sin x = 1/2

The system has been reduced to a simpler trigonometric equation.

Why These Methods Work

The expressions sin x and cos y can temporarily be treated as algebraic quantities.

For example, let:

a = sin x

b = cos y

Then:

a + b = 1

a − b = 0

This is an ordinary system of linear equations.

After solving for a and b, we return to the trigonometric equations to determine the angles.

Solving Systems by Elimination

Worked Example

Solve:

sin x + cos y = 1

sin x − cos y = 0

for:

0° ≤ x < 360°

0° ≤ y < 360°

Step 1: Add the Equations

sin x + cos y = 1

sin x − cos y = 0

Adding:

2sin x = 1

Therefore:

sin x = 1/2

Step 2: Solve for x

Sine is positive in Quadrants I and II.

Therefore:

x = 30°, 150°

Step 3: Find cos y

Substitute sin x = 1/2 into:

sin x − cos y = 0

Then:

1/2 − cos y = 0

Therefore:

cos y = 1/2

Cosine is positive in Quadrants I and IV.

Thus:

y = 60°, 300°

Step 4: Form All Ordered Pairs

Every permitted x-value can be paired with every permitted y-value because the reduced equations are independent.

The complete solution set is:

(30°, 60°)

(30°, 300°)

(150°, 60°)

(150°, 300°)

Step 5: Verify

For the pair (30°, 60°):

sin 30° + cos 60° = 1/2 + 1/2 = 1

sin 30° − cos 60° = 1/2 − 1/2 = 0

Both equations are satisfied.

The remaining pairs also satisfy both equations because they have the same required sine and cosine values.

Solving Systems by Substitution

Worked Example

Solve:

sin x + cos y = 1

sin x = 2cos y

for:

0° ≤ x < 360°

0° ≤ y < 360°

Step 1: Substitute

Replace sin x with 2cos y:

2cos y + cos y = 1

Therefore:

3cos y = 1

cos y = 1/3

Step 2: Solve for y

Let:

α = arccos(1/3)

α ≈ 70.53°

Cosine is positive in Quadrants I and IV.

Therefore:

y ≈ 70.53°, 289.47°

Step 3: Solve for x

From:

sin x = 2cos y

we obtain:

sin x = 2/3

Let:

β = arcsin(2/3)

β ≈ 41.81°

Sine is positive in Quadrants I and II.

Therefore:

x ≈ 41.81°, 138.19°

Step 4: Form the Solution Set

The four ordered pairs are approximately:

• (41.81°, 70.53°)

• (41.81°, 289.47°)

• (138.19°, 70.53°)

• (138.19°, 289.47°)

All four satisfy the system, subject to rounding.

Important Observation

The algebraic substitution produced one value for sin x and one value for cos y.

However, each trigonometric equation produced two possible angles.

This is why a system that appears to have one algebraic solution may have several trigonometric solutions.

Using Algebraic Variables to Simplify a System

A useful technique is to introduce temporary variables representing trigonometric expressions.

Worked Example

Solve:

2sin x + 3cos y = 2

sin x − cos y = 1/2

for:

0° ≤ x < 360°

0° ≤ y < 360°

Let:

a = sin x

b = cos y

The system becomes:

2a + 3b = 2

a − b = 1/2

Step 1: Isolate a

From the second equation:

a = b + 1/2

Step 2: Substitute

2(b + 1/2) + 3b = 2

Expand:

2b + 1 + 3b = 2

5b = 1

Therefore:

b = 1/5

Step 3: Find a

a = 1/5 + 1/2

a = 7/10

Therefore:

sin x = 0.7

cos y = 0.2

Step 4: Determine the Angles

For x:

x = arcsin(0.7)

or:

x = 180° − arcsin(0.7)

Approximately:

x ≈ 44.43°, 135.57°

For y:

y = arccos(0.2)

or:

y = 360° − arccos(0.2)

Approximately:

y ≈ 78.46°, 281.54°

Final Answer

The system has four solutions:

(44.43°, 78.46°)

(44.43°, 281.54°)

(135.57°, 78.46°)

(135.57°, 281.54°)

All values are approximate.

Recognising Impossible Systems

Not every algebraic solution corresponds to a valid trigonometric solution.

Recall:

−1 ≤ sin x ≤ 1

−1 ≤ cos x ≤ 1

Worked Example

Solve:

sin x + cos y = 3

sin x − cos y = 1

Add the equations:

2sin x = 4

sin x = 2

However, sine cannot equal 2 for any real angle.

Therefore:

The system has no real solutions.

General Principle

Always check that the values obtained for sine and cosine lie within their possible ranges.

Solving Systems with One Unknown Angle

Some simultaneous trigonometric equations involve only one variable.

In these cases, the solution must belong to the solution sets of both equations.

Worked Example

Solve:

sin x = 1/2

cos 2x = 1/2

for:

0° ≤ x < 360°

Step 1: Solve the First Equation

sin x = 1/2

Therefore:

x = 30°, 150°

Step 2: Check the Second Equation

For x = 30°:

cos 2x = cos 60° = 1/2

Valid.

For x = 150°:

cos 2x = cos 300° = 1/2

Valid.

Final Answer

x = 30°, 150°

Both values satisfy the complete system.

Alternative Method

Use:

cos 2x = 1 − 2sin² x

Since:

sin x = 1/2

then:

cos 2x = 1 − 2(1/2)²

= 1 − 1/2

= 1/2

Therefore, the second equation is automatically satisfied whenever the first equation is satisfied.

This reveals that the second equation imposes no additional restriction on the solutions of the first.

Solving Systems by Comparing Solution Sets

Worked Example

Solve:

sin x = cos x

sin 2x = 1

for:

0° ≤ x < 360°

Step 1: Solve the First Equation

sin x = cos x

Since cos x cannot equal zero at a solution, divide by cos x:

tan x = 1

Therefore:

x = 45°, 225°

Step 2: Solve the Second Equation

sin 2x = 1

2x = 90° + 360°n

Therefore:

x = 45° + 180°n

Within the interval:

x = 45°, 225°

Final Answer

x = 45°, 225°

Both equations produce the same solution set over the specified interval.

Systems with No Common Solutions

Worked Example

Solve:

sin x = 1/2

cos x = 1/2

for:

0° ≤ x < 360°

The first equation gives:

x = 30°, 150°

The second equation gives:

x = 60°, 300°

There are no common angles.

Therefore:

The system has no solution.

Algebraic Verification

If both equations were true, then:

sin² x + cos² x = (1/2)² + (1/2)²

= 1/4 + 1/4

= 1/2

But the Pythagorean identity requires:

sin² x + cos² x = 1

This contradiction confirms that no solution exists.

Systems Involving Sine and Cosine of the Same Angle

Systems containing sin x and cos x are especially interesting because the two expressions are not independent.

They must satisfy:

sin² x + cos² x = 1

Worked Example

Solve:

sin x + cos x = √2

sin x − cos x = 0

for:

0° ≤ x < 360°

Step 1: Add the Equations

2sin x = √2

Therefore:

sin x = √2/2

Step 2: Find cos x

From:

sin x − cos x = 0

we obtain:

cos x = √2/2

Step 3: Determine x

Both sine and cosine are positive in Quadrant I.

Therefore:

x = 45°

Verification

sin 45° + cos 45°

= √2/2 + √2/2

= √2

And:

sin 45° − cos 45° = 0

Both equations are satisfied.

Why Sine and Cosine of the Same Angle Cannot Be Chosen Independently

Suppose an algebraic system produces:

sin x = 0.8

cos x = 0.8

Both values lie individually within the interval from −1 to 1.

However:

sin² x + cos² x

= 0.64 + 0.64

= 1.28

This contradicts the Pythagorean identity.

Therefore, no real angle can satisfy both conditions.

Important Distinction

For a system involving sin x and cos y, the angles x and y are independent unless another equation relates them.

For a system involving sin x and cos x, the expressions are linked by the unit-circle identity.

This distinction determines whether separate angle solutions can be freely combined.

Solving Nonlinear Systems Using Trigonometric Identities

Some systems contain products or powers of trigonometric expressions.

Worked Example

Solve:

sin x + cos x = 1

sin x cos x = 0

for:

0° ≤ x < 360°

Step 1: Apply the Zero-Product Property

From:

sin x cos x = 0

we obtain:

sin x = 0

or:

cos x = 0

Case A: sin x = 0

The first equation becomes:

cos x = 1

Therefore:

x = 0°

Case B: cos x = 0

The first equation becomes:

sin x = 1

Therefore:

x = 90°

Final Answer

x = 0°, 90°

Verification

For x = 0°:

sin 0° + cos 0° = 0 + 1 = 1

sin 0° cos 0° = 0

For x = 90°:

sin 90° + cos 90° = 1 + 0 = 1

sin 90° cos 90° = 0

Both values satisfy the system.

Solving Systems Using Double-Angle Identities

Worked Example

Solve:

sin 2x = 0

cos x = −1

for:

0° ≤ x < 360°

Step 1: Solve the Second Equation

cos x = −1

Therefore:

x = 180°

Step 2: Verify the First Equation

sin 2(180°)

= sin 360°

= 0

Therefore:

x = 180°

is the only solution.

Alternative Method

Use:

sin 2x = 2sin x cos x

Then:

2sin x cos x = 0

Since cos x = −1, it is nonzero, so:

sin x = 0

Together with cos x = −1, this gives:

x = 180°

Systems with Two Unknown Angles and a Relationship Between Them

Some systems involve x and y through a direct relationship, such as:

x + y = 90°

This additional condition can significantly reduce the number of solutions.

Worked Example

Solve:

sin x = cos y

x + y = 90°

for:

0° ≤ x ≤ 90°

0° ≤ y ≤ 90°

Step 1: Use the Angle Relationship

From:

x + y = 90°

we obtain:

y = 90° − x

Step 2: Substitute

sin x = cos(90° − x)

Use the cofunction identity:

cos(90° − x) = sin x

Therefore:

sin x = sin x

This is an identity.

Interpretation

Every pair satisfying:

x + y = 90°

within the specified interval is a solution.

The complete solution set is:

(x, y) = (t, 90° − t)

where:

0° ≤ t ≤ 90°

Important Observation

This system has infinitely many solutions, even within the bounded interval.

The first equation does not provide any additional restriction beyond the second.

Solving Systems with a Sum of Angles

Worked Example

Solve:

sin x = cos y

x + y = 180°

for:

0° ≤ x < 180°

0° < y ≤ 180°

Step 1: Substitute

From:

x + y = 180°

we obtain:

y = 180° − x

Substitute:

sin x = cos(180° − x)

Use:

cos(180° − x) = −cos x

Therefore:

sin x = −cos x

Step 2: Solve

Since cos x cannot be zero at a solution:

tan x = −1

Within:

0° ≤ x < 180°

the solution is:

x = 135°

Step 3: Find y

y = 180° − 135°

y = 45°

Final Answer

(x, y) = (135°, 45°)

Verification

sin 135° = √2/2

cos 45° = √2/2

And:

135° + 45° = 180°

Both equations are satisfied.

Solving Systems with Quadratic Substitution

Some systems can be simplified by introducing:

a = sin x

b = cos y

and solving a nonlinear algebraic system.

Worked Example

Solve:

sin x + cos y = 1

sin x cos y = 3/16

for:

0° ≤ x < 360°

0° ≤ y < 360°

Step 1: Introduce Variables

Let:

a = sin x

b = cos y

Then:

a + b = 1

ab = 3/16

Step 2: Substitute

From:

a + b = 1

we obtain:

b = 1 − a

Substitute into:

ab = 3/16

Then:

a(1 − a) = 3/16

Expand:

a − a² = 3/16

Rearrange:

a² − a + 3/16 = 0

Multiply by 16:

16a² − 16a + 3 = 0

Factor:

(4a − 1)(4a − 3) = 0

Therefore:

a = 1/4

or:

a = 3/4

Step 3: Find b

If:

a = 1/4

then:

b = 3/4

If:

a = 3/4

then:

b = 1/4

Thus the possible trigonometric combinations are:

sin x = 1/4 and cos y = 3/4

or:

sin x = 3/4 and cos y = 1/4

Step 4: Determine the Angles

For sin x = 1/4:

x ≈ 14.48°, 165.52°

For cos y = 3/4:

y ≈ 41.41°, 318.59°

For sin x = 3/4:

x ≈ 48.59°, 131.41°

For cos y = 1/4:

y ≈ 75.52°, 284.48°

Final Answer

There are eight ordered pairs, formed by combining the two x-values with the two y-values within each valid algebraic case.

Each row produces four ordered pairs.

This example demonstrates how a nonlinear algebraic system can lead to several trigonometric solution sets.

General Solutions of Trigonometric Systems

When no interval is specified, solutions are often expressed using integer parameters.

Worked Example

Solve:

sin x = 1/2

cos y = √2/2

for all real x and y.

Step 1: General Solutions for x

sin x = 1/2

Therefore:

x = 30° + 360°m

or:

x = 150° + 360°m

Step 2: General Solutions for y

cos y = √2/2

Therefore:

y = 45° + 360°n

or:

y = 315° + 360°n

where:

m, n ∈ ℤ

Final Answer

The complete solution set consists of all combinations of the two x-families and two y-families.

The integers m and n are independent because x and y are independent variables.

Why Two Integer Parameters Are Needed

The value of x can repeat through one number of revolutions while y repeats through a different number.

Using the same integer for both variables would incorrectly exclude valid combinations.

Interpreting Multiple Solution Sets

A trigonometric system can have different types of solution sets.

Discrete and Continuous Solutions

A discrete solution set consists of separate individual solutions.

For example:

(30°, 60°)

(150°, 300°)

A continuous solution set contains an entire range of related solutions.

For example:

x + y = 90°

sin x = cos y

has the family:

(x, y) = (t, 90° − t)

for all permitted values of t.

Why This Matters

Two systems may both have infinitely many solutions but behave very differently.

One may have infinitely many isolated solutions due to periodic repetition.

Another may have infinitely many solutions forming a continuous curve.

Graphical Verification of Systems

Graphs provide a useful way to interpret and verify simultaneous trigonometric equations.

For equations involving one unknown, solutions are common intersection points.

For equations involving two unknowns, each equation defines a curve in the xy-plane. The simultaneous solutions occur where those curves intersect.

Example: One-Variable System

Consider:

sin x = cos x

sin 2x = 1

The first equation can be represented by the intersection of:

y = sin x

and:

y = cos x

The second equation can be represented by:

y = sin 2x

and:

y = 1

The valid solutions are the x-values that satisfy both relationships.

For:

0° ≤ x < 360°

the solutions are:

x = 45°, 225°

Graphical Interpretation

At x = 45°:

sin 45° = cos 45° = √2/2

sin 90° = 1

At x = 225°:

sin 225° = cos 225° = −√2/2

sin 450° = 1

Both points satisfy the system.

Graphing a Two-Variable System

Consider:

sin x + sin y = 1

sin x − sin y = 0

for:

0° ≤ x < 360°

0° ≤ y < 360°

Adding the equations gives:

2sin x = 1

Therefore:

sin x = 1/2

Subtracting gives:

2sin y = 1

Therefore:

sin y = 1/2

Thus:

x = 30°, 150°

y = 30°, 150°

The solutions are:

(30°, 30°)

(30°, 150°)

(150°, 30°)

(150°, 150°)

What Does the Graph Show?

The solution set contains four distinct ordered pairs.

Each point represents a combination of angles that satisfies both original equations.

The graph also demonstrates that finding two x-values and two y-values can produce four ordered pairs when the reduced conditions are independent.

Limitations of Graphical Verification

Graphs are useful for visual confirmation, but they may not reveal exact solutions.

For example, a graph may suggest:

x ≈ 41.8°

but an algebraic solution may be expressed exactly as:

x = arcsin(2/3)

Graphical methods should therefore support, rather than replace, algebraic reasoning.

Using Technology to Verify Solutions

Graphing calculators and mathematical software can help check trigonometric systems.

For a one-variable system, graph both equations and identify common solutions.

For a two-variable system, graph each equation as a relation between x and y.

Useful tools include:

• Desmos Graphing Calculator

• GeoGebra Graphing Calculator

Example Investigation

Enter:

y = sin x

y = cos x

Set the calculator to degree mode.

Observe the intersections over:

0° ≤ x < 360°

Then graph:

y = sin 2x

and:

y = 1

Compare the x-values satisfying both equations.

Important Reminder

A calculator must be set to the correct angle unit.

Entering angles in degrees while the calculator is in radian mode produces incorrect interpretations.

Practical Application: Resolving Forces

Simultaneous trigonometric equations can be used to determine the magnitudes of forces acting in different directions.

Suppose two forces act on an object.

Force A acts at 30° above the horizontal.

Force B acts at 60° above the horizontal.

Their horizontal and vertical components satisfy:

A cos 30° + B cos 60° = 100

A sin 30° + B sin 60° = 100

where A and B are measured in newtons.

Find the magnitudes of the two forces.

Step 1: Substitute Exact Values

Recall:

cos 30° = √3/2

sin 30° = 1/2

cos 60° = 1/2

sin 60° = √3/2

The equations become:

A√3/2 + B/2 = 100

A/2 + B√3/2 = 100

Multiply both equations by 2:

√3A + B = 200

A + √3B = 200

Step 2: Eliminate a Variable

From the first equation:

B = 200 − √3A

Substitute into the second:

A + √3(200 − √3A) = 200

Expand:

A + 200√3 − 3A = 200

Simplify:

−2A = 200 − 200√3

Therefore:

A = 100(√3 − 1)

Step 3: Find B

B = 200 − √3[100(√3 − 1)]

B = 200 − 300 + 100√3

Therefore:

B = 100(√3 − 1)

Final Answer

Both forces have the same magnitude:

A = B = 100(√3 − 1) N

Approximately:

A = B ≈ 73.21 N

Verification

Substituting A = B into the original equations gives the required horizontal and vertical totals of 100 N.

This application illustrates how trigonometric components can be combined with algebraic elimination.

Practical Application: Navigation

Suppose a boat travels along two separate paths.

The first displacement has magnitude a and direction 30° north of east.

The second displacement has magnitude b and direction 60° north of east.

The combined displacement is:

100 km east

and:

100 km north

The component equations are:

a cos 30° + b cos 60° = 100

a sin 30° + b sin 60° = 100

This is mathematically the same system as the previous force example.

Therefore:

a = b = 100(√3 − 1) km

Approximately:

a = b ≈ 73.21 km

Interpretation

The boat can achieve the specified displacement by travelling approximately 73.21 km in each of the two directions.

The same mathematical techniques apply to forces and navigation because both involve vector components.

Practical Application: Wave Interference

When two waves overlap, their displacements add.

Suppose two waves are represented by:

y₁ = A sin x

y₂ = A cos x

We want to find angles where their displacements are equal and their combined displacement is √2A.

The system is:

A sin x = A cos x

A sin x + A cos x = √2A

Assume A ≠ 0.

Divide both equations by A:

sin x = cos x

sin x + cos x = √2

Step 1: Use the First Equation

sin x = cos x

Step 2: Substitute into the Second

2sin x = √2

Therefore:

sin x = √2/2

Since sin x = cos x:

cos x = √2/2

Step 3: Determine x

Both functions are positive in Quadrant I.

Therefore:

x = 45° + 360°n

where n ∈ ℤ.

Interpretation

The two waves have equal positive displacements at these angles, producing the specified combined displacement.

Practical Application: Circular Motion

Suppose two objects move around a circular track.

Their vertical positions are:

y₁ = 5sin θ

y₂ = 5cos θ

Find the angles where the objects have equal vertical positions and both are above the horizontal axis.

Step 1: Set the Positions Equal

5sin θ = 5cos θ

Divide by 5:

sin θ = cos θ

Therefore:

tan θ = 1

The solutions within one revolution are:

θ = 45°, 225°

Step 2: Apply the Height Condition

At θ = 45°:

sin 45° = cos 45° = √2/2

Both positions are positive.

At θ = 225°:

sin 225° = cos 225° = −√2/2

Both positions are negative.

Therefore:

θ = 45°

is the only solution satisfying both conditions within one revolution.

Important Observation

A practical system may include inequalities or physical restrictions in addition to equations.

These restrictions help determine which mathematical solutions are meaningful.

A Systematic Strategy for Solving Trigonometric Systems

When solving simultaneous trigonometric equations, use the following process.

1. Identify the unknowns. Determine whether the system contains one angle or several.

2. Read the interval restrictions. Check the permitted values of each variable.

3. Examine the equation structure. Look for repeated expressions such as sin x or cos y.

4. Choose an algebraic method. Consider substitution, elimination, or introducing temporary variables.

5. Apply trigonometric identities when useful. Use Pythagorean, double-angle, or compound-angle identities to simplify the system.

6. Solve the resulting algebraic equations. Find the required trigonometric values.

7. Check possible ranges. Reject sine or cosine values outside −1 to 1.

8. Determine every angle. Use reference angles and quadrant rules.

9. Form ordered pairs carefully. Combine independent solutions only when permitted by the system.

10. Apply any additional relationships. Conditions such as x + y = 90° may restrict the pairs.

11. Remove duplicate solutions. Count each distinct solution once.

12. Verify every solution. Substitute into all original equations.

13. Use graphs when helpful. Confirm the number and approximate positions of solutions.

14. Interpret the results. Explain their meaning in practical contexts.

Common Mistakes

Solving only one equation

A valid solution must satisfy every equation in the system.

Finding only one trigonometric angle

A single sine or cosine value may correspond to two angles in one revolution.

Ignoring periodicity

General solutions may require integer multiples of 360° or 180°.

Combining dependent values incorrectly

sin x and cos x must satisfy the Pythagorean identity.

Failing to form all valid ordered pairs

Independent x- and y-solutions may produce several combinations.

Combining every possible pair without checking

Additional equations may restrict which combinations are valid.

Ignoring the range of sine and cosine

Values outside −1 to 1 cannot represent real sine or cosine values.

Using the wrong quadrant

Reference angles must be placed in quadrants with the correct trigonometric signs.

Mixing degrees and radians

Angle units must remain consistent.

Forgetting interval restrictions

Only solutions within the required intervals should be included.

Dividing by expressions that may equal zero

This can eliminate valid solutions.

Ignoring physical restrictions

Negative force magnitudes or impossible times may be inappropriate in practical contexts.

Assuming a graph proves an exact solution

Graphical methods usually provide approximate verification.

Failing to check both original equations

Every proposed solution must satisfy the complete system.

Rounding too early

Keep exact values until the final calculation when possible.

Key Terms

System of Trigonometric Equations: Two or more trigonometric equations that must be satisfied simultaneously.

Simultaneous Equations: Equations involving shared unknowns that are solved together.

Solution Set: The complete collection of values satisfying all equations.

Ordered Pair: A pair of values (x, y) in which the order identifies the value of each variable.

Substitution Method: An algebraic method in which an expression from one equation replaces an equivalent expression in another.

Elimination Method: An algebraic method that combines equations to remove an unknown or repeated expression.

Temporary Variable: A new variable introduced to simplify an algebraic or trigonometric expression.

Linear System: A system in which the unknown quantities occur to the first power and are not multiplied together.

Nonlinear System: A system containing products, powers, or other nonlinear relationships involving unknown quantities.

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

Pythagorean Identity: The relationship sin² x + cos² x = 1.

Double-Angle Identity: An identity involving trigonometric functions of twice an angle.

Compound-Angle Identity: An identity involving the sum or difference of angles.

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

Quadrant: One of the four regions of the Cartesian coordinate plane.

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

General Solution: A formula representing all solutions using integer parameters.

Independent Variables: Variables that can vary separately within the restrictions of a system.

Dependent Variables: Variables linked by an equation or other mathematical relationship.

Discrete Solution Set: A collection of separate individual solutions.

Continuous Solution Set: A family of solutions containing an uninterrupted range of values.

Graphical Verification: Checking solutions using intersections or relationships between graphs.

Intersection Point: A point where two curves meet.

Domain Restriction: A condition limiting the permitted input values of an expression.

Extraneous Solution: A candidate obtained during manipulation that does not satisfy the original system.

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

Factorisation: Rewriting an expression as a product of factors.

Vector Component: The part of a vector acting along a specified direction.

Resultant Vector: A vector representing the combined effect of two or more vectors.

Wave Interference: The combination of two or more waves.

Circular Motion: Motion along a circular path.

Mathematical Modelling: The use of mathematical equations to represent real-world situations.

Verification: Checking that a proposed solution satisfies every original equation.

Check Your Understanding

1. Explain the difference between solving one trigonometric equation and solving a system of trigonometric equations.

2. Describe how substitution and elimination can be used to solve trigonometric systems.

3. Solve:

sin x + cos y = 1

sin x − cos y = 0

for 0° ≤ x < 360° and 0° ≤ y < 360°.

4. Solve:

sin x + sin y = 1

sin x − sin y = 0

for 0° ≤ x < 360° and 0° ≤ y < 360°.

5. Solve:

2sin x + cos y = 1

sin x − cos y = 0

for 0° ≤ x < 360° and 0° ≤ y < 360°.

6. Explain why the system:

sin x = 0.8

cos x = 0.8

has no real solution.

7. Solve:

sin x = 1/2

cos 2x = 1/2

for 0° ≤ x < 360°.

8. Solve:

sin x = cos x

sin 2x = 1

for 0° ≤ x < 360°.

9. Solve:

sin x + cos x = √2

sin x − cos x = 0

for 0° ≤ x < 360°.

10. Solve:

sin x + cos x = 1

sin x cos x = 0

for 0° ≤ x < 360°.

11. Solve:

sin x = cos y

x + y = 180°

for 0° ≤ x < 180° and 0° < y ≤ 180°.

12. Solve:

sin x + cos y = 1

sin x cos y = 3/16

for 0° ≤ x < 360° and 0° ≤ y < 360°.

13. Determine the general solutions of:

sin x = √3/2

cos y = −1/2

where x and y are measured in degrees.

14. Explain how graphs can be used to verify the solution set of a simultaneous trigonometric system. Describe one limitation of graphical verification.

15. Two forces A and B act at angles of 30° and 60° above the horizontal. Their resultant has a horizontal component of 80 N and a vertical component of 100 N.

Construct two simultaneous equations using horizontal and vertical components. Solve for A and B, verify your results, and explain the physical meaning of your answers.

Key Takeaways

• A system of trigonometric equations requires every equation to be satisfied simultaneously.

• Systems may involve one unknown angle or several unknown angles.

• Algebraic substitution and elimination are useful methods for solving trigonometric systems.

• Temporary variables can simplify systems containing repeated trigonometric expressions.

• A single sine or cosine value may correspond to multiple angles.

• Systems involving two independent angles may produce several ordered pairs.

• The order of values in an ordered pair matters.

• Independent trigonometric solutions can be combined when no additional relationship restricts them.

• Sine and cosine of the same angle are linked by sin² x + cos² x = 1.

• Algebraic solutions must be checked against the ranges of trigonometric functions.

• Some systems have one solution, several solutions, infinitely many solutions, or no solutions.

• General solutions use integer parameters to represent periodic repetition.

• Additional angle relationships can restrict the possible solution pairs.

• Trigonometric identities help simplify nonlinear systems.

• Double-angle identities are useful when a system contains both x and 2x.

• Graphical verification helps identify intersections and confirm solution sets.

• Graphs are generally less precise than exact algebraic solutions.

• Every proposed solution must be substituted into all original equations.

• Practical restrictions may eliminate mathematically valid but physically inappropriate solutions.

• Systems of trigonometric equations are useful in force analysis, navigation, circular motion, and wave interference.

• Mastering simultaneous trigonometric equations develops algebraic fluency, logical reasoning, graphical interpretation, and mathematical modelling skills essential for advanced mathematics, physics, and engineering.