Solve trigonometric equations on a specified interval

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

Find every solution in the stated domain using exact methods and graphing technology

Trigonometric equations can have multiple solutions because sine, cosine and tangent repeat their values. The specified interval determines which solutions to include. Before solving, identify whether angles are in radians or degrees, check whether the endpoints are included, and set a calculator to the correct angle mode. This lesson uses radians.

What you will learn

Solve sine and cosine equations on a specified interval using exact angles where possible.
Use periodicity and interval endpoints to find all solutions, including equations with a multiple angle.
Use graphs and numerical methods to estimate and check solutions, reporting approximations to an appropriate accuracy.

1. Prior knowledge: angles, periods and intervals

An interval describes the allowed values of a variable. The notation

[0,2\pi)

includes

0

but excludes

2\pi

. The notation

[-\pi,2\pi]

includes both endpoints. A solution at an included endpoint counts if it satisfies the equation.

Sine and cosine repeat after a full turn, while tangent repeats after half a turn. These repeating lengths are called periods. Periodicity lets us describe all matching angles, then select the ones in the required interval.

An inverse-trigonometric button gives a principal angle, not necessarily every angle with a given trigonometric value. For example,

\sin^{-1}(1/2)=\pi/6

gives one angle. Other angles can have the same sine, so use known angle patterns or a graph to find all solutions.

\sin(x+2\pi)=\sin x,\quad \cos(x+2\pi)=\cos x,\quad \tan(x+\pi)=\tan x

Read the interval brackets before solving; they determine whether endpoints count.
Use one angle unit consistently. The intervals and examples here use radians.
Sine and cosine have period $2\pi$ ; tangent has period $\pi$ .

2. A reliable method for finding all solutions

First use algebra to isolate the trigonometric function. For example, rearranging

3\cos x-1=0

gives a specific value for

\cos x

. Do not use an inverse-trigonometric button until the trigonometric function is isolated.

Next find every angle that gives the required value. For exact familiar values, use known angles and the function’s pattern over a full period. For other values, an inverse-trigonometric result can provide a starting angle; then use a graph or the function’s pattern to find any other angles.

If the equation contains an inside angle such as

2x

or

3x

, transform the interval to describe that inside angle. For example, if

0\leq x<2\pi

, then

0\leq 3x<6\pi

. Find the allowed inside-angle values first, then divide by the coefficient to obtain values of

x

.

Finally, keep only solutions in the original interval and check them in the original equation. This process avoids missing repeated solutions or including values outside the domain.

0\leq x<2\pi\;\Longrightarrow\;0\leq 3x<6\pi

Isolate the trigonometric function before finding angles.
Apply interval bounds to the complete angle inside the function.
Check every candidate in the original equation and against the interval.

3. Graphical and numerical representations

An equation such as

\sin x=0.4

can be read graphically as the intersections of

y=\sin x

and

y=0.4

. Each intersection’s horizontal coordinate is a solution. The algebraic method identifies why the angles match; the graph helps show how many intersections occur in the interval.

For a more complicated equation, such as

2\sin x=x

, graph

y=2\sin x

and

y=x

on the specified interval. A graphing calculator’s intersection or numerical-solver feature can estimate where the curves meet. It may report only one intersection, so inspect the whole interval and search for any others.

For a useful technology check, set the calculator to radians when the interval is in radians, choose a viewing window that covers the entire interval, and zoom near intersections if needed. Substitute an unrounded calculator value into the original equation to check it. Then report the result to the requested accuracy.

In a context, the variable might represent time or an angle. The interval may reflect the allowed times or angles, and units should be stated. A mathematical solution outside that contextual interval is not an appropriate answer.

Graph intersections represent solutions, but the full interval must be checked.
Use a calculator to estimate and verify numerical solutions, not as a replacement for reasoning.
Keep track of angle units and any context-specific restrictions.

Worked example

Exact solutions with a multiple angle

Solve

\sin(2x)=\frac{\sqrt{3}}{2}

for

0\leq x<2\pi

.

Find the range of the inside angle
Since $x$ ranges from $0$ up to, but not including, $2\pi$ , doubling the bounds gives the full range for $2x$ . This range covers two full sine periods.
$0\leq 2x<4\pi$
Find the matching sine angles
In one full turn, sine is $\frac{\sqrt{3}}{2}$ at $\frac{\pi}{3}$ and $\frac{2\pi}{3}$ . Repeating both angles by periods of $2\pi$ gives all matching angles in the range for $2x$ . The integer $k$ counts the full turns in these patterns. 2x= $\frac{\pi}{3}$ +2k\pi\quador 2x= $\frac{2\pi}{3}$ +2k\pi, k∈ $\mathbb{Z}$
Divide and select the interval values
Divide the matching angles by $2$ , then retain only values of $x$ with $0\leq x<2\pi$ . This gives two solutions from each of the two angle patterns.
$x=\frac{\pi}{6},\;\frac{\pi}{3},\;\frac{7\pi}{6},\;\frac{4\pi}{3}$

Answer:

x=\frac{\pi}{6},\frac{\pi}{3},\frac{7\pi}{6},\frac{4\pi}{3}

Check: Doubling the four values gives

\frac{\pi}{3},\frac{2\pi}{3},\frac{7\pi}{3},\frac{8\pi}{3}

. Each lies in

[0,4\pi)

and has sine

\frac{\sqrt{3}}{2}

.

Worked example

Cosine solutions on an interval crossing zero

Solve

2\cos x-1=0

for

-\pi\leq x\leq 2\pi

.

Isolate cosine
Rearrange the equation using ordinary algebra. The resulting cosine value is one of the familiar exact values.
$\cos x=\frac{1}{2}$
Write the repeating angle patterns
Cosine is $\frac{1}{2}$ at $\frac{\pi}{3}$ and $-\frac{\pi}{3}$ within a full turn. Adding full turns gives all candidates for any interval. The integer $k$ counts the full turns in these patterns. x= $\frac{\pi}{3}$ +2k\pi\quador x=- $\frac{\pi}{3}$ +2k\pi, k∈ $\mathbb{Z}$
Select values in the stated interval
Check the candidates against both interval bounds. The values between $-\pi$ and $2\pi$ , including the endpoints if applicable, are $-\frac{\pi}{3}$ , $\frac{\pi}{3}$ , and $\frac{5\pi}{3}$ .
$-\pi\leq x\leq 2\pi$

Answer:

x=-\frac{\pi}{3},\frac{\pi}{3},\frac{5\pi}{3}

Check: Cosine is

\frac{1}{2}

at each listed angle. Neither endpoint is a solution.

Worked example

A numerical solution checked by graphing

Solve

2\sin x=x

for

0\leq x\leq\pi

. Give answers to three significant figures.

Interpret the equation as an intersection
In radian mode, graph $y=2\sin x$ and $y=x$ over the entire interval $[0,\pi]$ . A point where the graphs meet has an $x$ -coordinate that solves the equation.
$2\sin x=x$
Locate and estimate the intersections
The graphs meet at $x=0$ . A second intersection is between $1.8$ and $2.0$ radians. A graphing calculator’s intersection or numerical-solver feature refines the second value to approximately $1.89549$ .
$x\approx 1.89549$
Report the requested accuracy
The interval includes both intersections. To three significant figures, the second value is $1.90$ radians; the exact endpoint solution is $0$ . x=0\quador x\approx 1.90

Answer:

x=0

or

x\approx 1.90

radians, to three significant figures.

Check: At

x=0

, both sides are zero. Substituting the unrounded numerical estimate gives

2\sin(1.89549)\approx1.89549

, which supports the second solution.

Common mistakes and how to avoid them

Using only the angle returned by an inverse-trigonometric button.

Correction: Find all angles with the required value in the stated interval, using the graph or the repeating angle pattern.

Applying the interval for

x

directly to an inside angle such as

2x

.

Correction: Transform both interval bounds to describe the inside angle before selecting solutions.

Including a value outside the interval or overlooking an included endpoint.

Correction: Test each candidate against the interval notation and the original equation.

Assuming a calculator’s single reported root is the complete answer.

Correction: Inspect the graph over the entire interval for all intersections, then check each candidate.

Lesson summary

Isolate the trigonometric function and identify every matching angle.
Use periodicity and the specified interval to select all allowed solutions.
For a multiple angle, first find the range of the angle inside the function.
Use graphs and numerical solvers to check solutions, and state the accuracy of approximations.

Check your understanding

Question 1

How many solutions does

\cos x=0

have on

[0,2\pi)

?

One: $x=\frac{\pi}{2}$
Two: $x=\frac{\pi}{2},\frac{3\pi}{2}$
Three: $x=0,\frac{\pi}{2},\frac{3\pi}{2}$
Four: $x=\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi$

Show answer and explanation

Two:

x=\frac{\pi}{2},\frac{3\pi}{2}

Cosine is zero at

\frac{\pi}{2}

and

\frac{3\pi}{2}

during one full turn. The other listed values are not zeros; in addition,

2\pi

is excluded.

Question 2

If

0\leq x<2\pi

, what range does

3x

cover?

$0\leq 3x<2\pi$
$0\leq 3x<3\pi$
$0\leq 3x<6\pi$
$0<3x\leq 6\pi$

Show answer and explanation

0\leq 3x<6\pi

Multiply both bounds by the positive number

3

, so the inequality directions stay the same and the upper endpoint remains excluded.

Question 3

Which calculator setup is appropriate when solving on

[0,\pi]

?

Degree mode, because the interval has two endpoints
Radian mode, with a viewing window covering $[0,\pi]$
Either mode, without changing the interval
Radian mode, with a viewing window covering only $[0,1]$

Show answer and explanation

Radian mode, with a viewing window covering

[0,\pi]

The interval is expressed in radians. Set radian mode and display the whole interval so that all possible intersections can be examined.

Key terms

Specified interval: The stated range of allowed values for the variable, including the endpoint rules shown by brackets or parentheses.
Period: The positive angle by which a trigonometric function can be shifted without changing its values.
Intersection: A point where two graphs meet; its horizontal coordinate can solve an equation formed by setting the functions equal.
Principal angle: The angle returned by an inverse-trigonometric function on a calculator; other angles can have the same trigonometric value.

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 3.8. It is a study resource, not an official curriculum publication.

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