Trigonometric Equations

The Central Idea

A trigonometric equation is an equation containing one or more trigonometric functions of an unknown angle.

For example,

asks for every angle \(x\) whose sine equals \(1/2\).

Because trigonometric functions are periodic, such equations often have infinitely many solutions.

Identity versus Equation

A trigonometric identity is true throughout its common domain.

A trigonometric equation is true only for selected inputs.

For example,

is an identity, while

is an equation to solve.

What Counts as a Solution?

A solution is an angle that makes the original equation true.

A candidate is not accepted merely because it appears during algebraic manipulation.

The final check is always substitution into the original equation.

Solutions on an Interval

Some problems restrict the answer to an interval such as

In that case, list only the solutions inside the specified interval.

Periodic repetitions outside the interval are not included.

General Solutions

If no finite interval is specified, the goal is often to describe every real solution.

Periodicity makes this possible using integer parameters such as

For example,

has the general solution

Start by Isolating the Trigonometric Function

Many equations become familiar after ordinary algebra isolates the trigonometric expression.

For example,

becomes

The remaining work is then trigonometric rather than algebraic.

Solving \(\sin x=a\)

If

first identify a reference or principal angle.

Then use the signs of sine to locate all relevant quadrants.

Sine is positive in Quadrants I and II and negative in Quadrants III and IV.

Example: Solve \(\sin x=1/2\) on One Revolution

Solve

for

The reference angle is

Sine is positive in Quadrants I and II.

Therefore,

General Sine Solutions

If

then the two basic solution families are

and

where

General Solution for \(\sin x=1/2\)

From the one-revolution solutions,

the complete solution is

or

Solving \(\cos x=a\)

Cosine is positive in Quadrants I and IV and negative in Quadrants II and III.

Once a reference angle is known, use the sign of the requested cosine value to determine the appropriate quadrants.

Example: Solve \(\cos x=-\sqrt2/2\)

Solve on

The reference angle is

Cosine is negative in Quadrants II and III.

Therefore,

General Cosine Solutions

If

then

where

This compact form expresses the reflection symmetry of cosine.

Solving \(\tan x=a\)

Tangent has period

so once one solution \(\alpha\) is known, all solutions are

Tangent is positive in Quadrants I and III and negative in Quadrants II and IV.

Example: Solve \(\tan x=1\)

The principal/reference angle is

Tangent repeats every \(\pi\).

Therefore,

Example: Tangent on One Revolution

Solve

for

The reference angle is

Tangent is negative in Quadrants II and IV.

Therefore,

Inverse Trigonometric Functions Give Principal Values

When an exact unit-circle value is unavailable, use an inverse trigonometric function to obtain a principal angle.

For example,

gives

This is only one principal value. The complete solution must still account for sine's symmetry and period.

Example: Non-Special Sine Value

Solve

for

The principal angle is approximately

Since sine is positive in Quadrants I and II, the second solution is

Thus,

Equations with Reciprocal Functions

Reciprocal equations can often be rewritten in terms of sine, cosine, or tangent.

For example,

is equivalent to

Then solve the familiar cosine equation.

Example: Solve \(\csc x=-2\)

Rewrite:

On

sine is negative in Quadrants III and IV, with reference angle \(\pi/6\).

Therefore,

Example: Solve \(\cot x=-1\)

Since

the equation is equivalent to

Thus,

Use Algebra before Trigonometry

Consider

First isolate cosine:

Only then use inverse cosine and quadrant reasoning.

Quadratic-Looking Trigonometric Equations

Some equations are quadratic in a trigonometric function.

For example,

has the same algebraic structure as

if

Example: Factor a Trigonometric Quadratic

Solve

Factor:

Therefore,

or

Complete the Factored Example

On

the equation

gives

The equation

gives

Thus,

Zero-Product Reasoning

If an equation has been written as

then at least one factor must be zero:

or

This familiar algebraic principle is extremely useful in trigonometric equations.

Example: Factor a Common Trigonometric Factor

Solve

Factor:

Therefore,

or

Do Not Divide Away Possible Solutions

In

dividing both sides by \(\sin x\) would produce

But this division assumes

Any solutions with \(\sin x=0\) would be lost.

Factoring is safer.

Using Trigonometric Identities

Identities can convert an equation into a more solvable form.

The most common substitutions include

and

Example: Use the Pythagorean Identity

Solve

Replace

with

Then

Continue the Identity Example

Subtract \(1\):

Multiply by \(-1\):

Move all terms to one side:

Factor:

Finish the Identity Example

Thus,

or

On

the solutions are

Equations Involving Both Sine and Cosine

If an equation contains both sine and cosine, possible strategies include:

• factoring;
• using a Pythagorean identity;
• dividing only after separately checking zero cases;
• rewriting tangent as \(\sin x/\cos x\);
• rewriting everything in one trigonometric function.

Example: Convert to Tangent

Suppose

and \(\cos x\neq0\).

Divide by \(\cos x\):

Therefore,

Hence,

Why the Division Was Safe in That Example

If

the original equation

would require

at the same angle, which never occurs.

Therefore no valid solutions are lost by dividing by \(\cos x\) in this particular equation.

Equations with a Transformed Angle

The trigonometric input may itself contain an algebraic expression:

A useful strategy is to temporarily treat the entire argument as one angle.

Let

Then solve

before converting back to \(x\).

Example: Solve \(\sin(2x)=1/2\)

Let

Then

Thus,

or

Convert Back to \(x\)

Since \(u=2x\),

gives

The second family gives

Therefore,

Interval Restrictions with Transformed Arguments

Suppose

must be solved for

Then

The doubled argument covers two full sine cycles, so more solutions appear than in the basic equation \(\sin x=1/2\).

Solutions of the Transformed Example on \(0\leq x<2\pi\)

From

and

choose integer values giving \(0\leq x<2\pi\):

Equations with a Phase Shift

Consider

Let

Since

when

we obtain

Equations with a Negative Argument

Even–odd identities can simplify signs.

Since

the equation

becomes

Equations That Have No Real Solutions

Sine and cosine outputs must lie between \(-1\) and \(1\).

Therefore,

and

have no real solutions.

Always check the possible range before doing unnecessary work.

Boundary Values

Equations such as

have only one solution per sine period:

Similarly,

gives

Special Zero Equations

Several common zero equations have compact general forms:

and

Domain Restrictions in Equations

An equation may contain expressions that are undefined at certain angles.

For example,

requires

A candidate where the original expression is undefined can never be a solution.

Extraneous Solutions

An extraneous solution is a candidate produced by a transformation that does not satisfy the original equation.

Trigonometric equations can acquire extraneous candidates through:

• squaring both sides;
• multiplying by expressions that may be zero;
• changing domains through simplification;
• using algebraically nonreversible steps.

Example: Squaring Can Create Candidates

Suppose

Squaring gives

But the squared equation is also true when

Squaring has therefore weakened the original sign condition.

Any solutions found after squaring must be checked in the original equation.

Graphical Interpretation of a Trigonometric Equation

The equation

can be interpreted as asking where the graphs

and

intersect.

This explains visually why multiple solutions occur.

Graphing as a Check

A graph can help:

• estimate the number of solutions on an interval;
• check approximate locations;
• identify missed periodic repetitions;
• detect impossible answers.

Exact algebraic and unit-circle reasoning should still be used when exact solutions are required.

Numerical Solutions

Not every trigonometric equation has elementary exact solutions.

For example,

normally leads to decimal approximations unless an exact inverse form is acceptable.

In such cases, state the requested precision clearly.

Inverse Form Can Be Exact

Even when no familiar special angle exists, the expression

is an exact symbolic description of the principal angle.

A decimal is an approximation to that exact value.

Periodicity Determines the Final Pattern

The period of the function determines how the solution families repeat.

Sine and cosine use shifts by

Tangent and cotangent use shifts by

When the argument contains \(Bx\), the corresponding period in \(x\) is compressed or stretched.

General Solution with a Linear Argument

Consider

Since

gives

we have

Therefore,

Factoring a Difference of Squares

Consider

Factor:

Thus,

or

Complete the Difference-of-Squares Example

The first equation gives

while the second gives

Together, the solutions may be expressed as

Solving with a Substitution

For

let

Then

Factor:

Therefore,

Complete the Substitution Example

On

cosine equals \(1/2\) in Quadrants I and IV:

When a Quadratic Root Is Impossible

Suppose algebra produces

Since

that algebraic branch contributes no real trigonometric solutions.

Range restrictions can eliminate roots immediately.

Exact Solutions Should Stay Exact

If a solution is

do not replace it unnecessarily with a decimal.

Exact radian forms preserve the unit-circle structure and make periodic generalizations easier to see.

Degree-Measure Equations

The same methods work in degrees.

For example,

has solutions

and

during one revolution.

The general solutions repeat by

General Degree Solutions

For sine,

gives

or

Tangent repeats every

Calculator Mode Matters

If the equation is being solved in radians, inverse-trigonometric calculations should be performed in radian mode.

If the expected answers are in degrees, use degree mode or convert explicitly.

A mode mismatch can produce a numerically plausible but contextually wrong answer.

Verify after Approximation

If a solution is rounded, substitute it into the original equation and expect approximate rather than exact equality.

The residual difference should be consistent with the stated rounding.

Common Error: Returning Only the Inverse-Function Output

If

writing only

misses the Quadrant-II solution and all periodic repetitions.

Common Error: Using \(2\pi\) for Tangent's Repetition

Tangent repeats after

not merely after \(2\pi\).

Therefore,

usually has a general solution of the form

Common Error: Forgetting the Given Interval

A general solution may be mathematically correct but fail the requested format if the problem asks only for

Always read the domain or interval before presenting the final answer.

Common Error: Dividing by a Trigonometric Expression Too Soon

Dividing by

can remove solutions for which

Before dividing by a variable expression, either factor first or separately analyze the zero case.

Common Error: Ignoring Function Range

Equations such as

have no real solutions.

Recognizing the range

can settle the equation immediately.

Common Error: Forgetting to Divide the Entire Angle Equation

If

dividing by \(2\) gives

The periodic term must be divided as well.

Common Error: Assuming Squaring Is Reversible

Squaring removes sign information.

Therefore candidates produced after squaring must be checked in the original unsquared equation.

A Reliable Procedure for Basic Trigonometric Equations

1. Identify the requested interval or whether a general solution is required.
2. Use ordinary algebra to isolate the trigonometric function.
3. Check whether the requested value lies in the function's range.
4. Find a reference or principal angle.
5. Use quadrant signs to find all solutions in one period.
6. Apply periodicity to construct the general solution if needed.
7. Restrict the solutions to the requested interval.
8. Verify in the original equation.

A Reliable Procedure for Factorable Equations

1. Move all terms to one side.
2. Factor algebraically.
3. Apply the zero-product property.
4. Solve each resulting trigonometric equation separately.
5. Combine the solution sets.
6. Remove duplicates.
7. Check all solutions against the original equation and interval.

A Reliable Procedure for Equations Using Identities

1. Identify which trigonometric functions appear.
2. Choose an identity that rewrites the equation in fewer function types.
3. Simplify using ordinary algebra.
4. Factor or substitute if the result is quadratic-like.
5. Solve the resulting basic trigonometric equations.
6. Preserve all domain restrictions introduced by the original expression.
7. Verify every candidate.

A Reliable Procedure for Transformed Arguments

1. Treat the complete argument, such as \(2x-\pi/3\), as one temporary angle.
2. Solve the basic trigonometric equation for that angle.
3. Write the complete periodic families before solving back for \(x\).
4. Undo the algebraic transformation.
5. Apply any interval restriction to \(x\).
6. Verify representative or all interval solutions.

Practice 1: Basic Sine Equation

Solve

for

Practice 2: Basic Cosine Equation

Solve

for

Practice 3: General Tangent Equation

Solve

for all real \(x\).

Practice 4: Algebra before Trigonometry

Solve

on

Practice 5: Reciprocal Function

Solve

on

Practice 6: Factorable Equation

Solve

on

Practice 7: Trigonometric Quadratic

Solve

on

Practice 8: Use an Identity

Solve

for all real \(x\).

Practice 9: Transformed Argument

Solve

on

Practice 10: Phase Shift

Solve

for all real \(x\).

Practice 11: No Real Solution

Solve

over the real numbers.

Practice 12: Identify the Error

A student solves

by dividing both sides by \(\sin x\) and concludes that

gives every solution.

Explain the error.

Practice Answers

Practice 1

The reference angle is

Sine is positive in Quadrants I and II:

Practice 2

The reference angle is

Cosine is negative in Quadrants II and III:

Practice 3

A principal solution is

Tangent repeats every \(\pi\):

Practice 4

gives

Therefore,

Practice 5

means

Thus,

Practice 6

Use the zero-product property:

or

On \(0\leq x<2\pi\),

gives

The second equation gives \(x=\pi\), already included.

Therefore,

Practice 7

Factor:

Thus,

or

Therefore,

Practice 8

Rewrite as

Factor:

This gives tangent values \(1\) and \(-1\), so

Practice 9

Cosine is zero when

Therefore,

Restricting to \(0\leq x<2\pi\) gives

Practice 10

Sine is zero when its argument is \(k\pi\):

Therefore,

Practice 11

requires

But sine cannot exceed \(1\).

Therefore,

Practice 12

Dividing by \(\sin x\) assumes

But any angle satisfying

already makes the original product equal to zero.

Those solutions are lost by the division.

The correct approach is to keep the factorization:

or

What to Remember

• A trigonometric equation is true only for selected angles; an identity is true throughout its common domain.
• First isolate or simplify the trigonometric expression using ordinary algebra.
• Use unit-circle values and reference angles for exact solutions.
• Inverse trigonometric functions return principal values, not automatically every solution.
• Sine and cosine repeat every \(2\pi\); tangent and cotangent repeat every \(\pi\).
• Quadrant signs determine the additional solutions associated with a reference angle.
• Factoring and the zero-product property are often safer than dividing by a trigonometric expression.
• Identities can rewrite equations into a single trigonometric function.
• For transformed inputs such as \(2x-\pi/3\), solve for the entire angle first and then solve back for \(x\).
• Check the range of sine and cosine before attempting impossible equations.
• Squaring and other nonreversible steps may introduce extraneous candidates.
• Always respect the requested interval and verify candidates in the original equation.

Where We Go Next

Trigonometric equations combine algebra, identities, inverse functions, unit-circle geometry, and periodicity into one problem-solving framework.

These same ideas recur whenever periodic models must be inverted, crossing times must be found, phase conditions must be solved, or trigonometric relationships are used in later calculus and applied mathematics.