4. Solving Right Triangles

What Does It Mean to Solve a Right Triangle?

To solve a triangle means to determine every unknown side length and every unknown angle.

Every triangle has:

• three sides
• three interior angles

A right triangle already contains one known angle:

90°

Therefore, solving a right triangle usually means finding:

• any unknown side lengths
• the two acute angles

A complete solution gives enough information to describe the entire triangle.

---

The Parts of a Right Triangle

A right triangle contains one 90° angle.

Relative to one of the other angles, the sides are called:

• opposite
• adjacent
• hypotenuse

The hypotenuse is always opposite the 90° angle and is always the longest side.

The terms opposite and adjacent depend on which acute angle is being considered.

---

Information Needed to Solve a Right Triangle

To solve a right triangle, you normally need at least two additional pieces of information besides the 90° angle.

For example:

• two sides
• one side and one acute angle

From these measurements, the remaining sides and angles can usually be calculated.

---

The Main Tools

Several mathematical tools can be used when solving right triangles:

Pythagorean Theorem

Used when two sides are known and the third side is required.

Sine

Used with the opposite side and hypotenuse.

Cosine

Used with the adjacent side and hypotenuse.

Tangent

Used with the opposite and adjacent sides.

Inverse trigonometric functions

Used when side lengths are known and an angle must be found.

Angle sum

Used to find the remaining acute angle.

The important skill is deciding which tool is most efficient for the information available.

---

The Pythagorean Theorem

When two sides of a right triangle are known, the third can often be calculated using the Pythagorean theorem.

Here:

a and b are the shorter sides.

c is the hypotenuse.

---

Example 1: Finding the Hypotenuse

Suppose the shorter sides are:

a = 6 cm

b = 8 cm

Then:

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = √100

c = 10 cm

The hypotenuse is 10 cm.

This is the well-known 6–8–10 right triangle.

---

Finding a Shorter Side

If the hypotenuse and one shorter side are known, rearrange the Pythagorean theorem.

Suppose:

c = 13 cm

a = 5 cm

Find b.

5² + b² = 13²

25 + b² = 169

b² = 144

b = 12 cm

The missing side is 12 cm.

---

SOH CAH TOA

When an angle and at least one side are involved, trigonometric ratios are often useful.

SOH

sin θ = opposite ÷ hypotenuse

CAH

cos θ = adjacent ÷ hypotenuse

TOA

tan θ = opposite ÷ adjacent

Remember:

SOH – CAH – TOA

---

Selecting the Correct Trigonometric Method

Ask:

Which sides do I know or need?

If the problem involves:

Opposite + Hypotenuse → sine

Adjacent + Hypotenuse → cosine

Opposite + Adjacent → tangent

Do not select a trig function based on habit.

Select it based on the sides involved.

---

Example 2: Finding a Side Using Sine

A right triangle has:

Angle = 35°

Hypotenuse = 12 cm

Find the side opposite the 35° angle.

Use sine:

sin 35° = opposite ÷ 12

Let the opposite side be x.

sin 35° = x ÷ 12

x = 12 sin 35°

x ≈ 6.88 cm

The opposite side is approximately 6.88 cm.

---

Example 3: Finding a Side Using Cosine

A right triangle has:

Angle = 42°

Hypotenuse = 18 m

Find the adjacent side.

cos 42° = adjacent ÷ 18

Let the adjacent side be x.

cos 42° = x ÷ 18

x = 18 cos 42°

x ≈ 13.38 m

The adjacent side is approximately 13.4 m.

---

Example 4: Finding a Side Using Tangent

A right triangle has:

Angle = 28°

Adjacent side = 15 cm

Find the opposite side.

tan 28° = opposite ÷ 15

Let the opposite side be x.

tan 28° = x ÷ 15

x = 15 tan 28°

x ≈ 7.98 cm

The opposite side is approximately 7.98 cm.

---

Finding Angles

When side lengths are known but an angle is unknown, use an inverse trigonometric function.

These are:

sin⁻¹

cos⁻¹

tan⁻¹

These functions allow us to work backward from a side ratio to an angle.

---

Example 5: Finding an Angle Using Tangent

Suppose:

Opposite = 7 cm

Adjacent = 10 cm

Find θ.

tan θ = 7 ÷ 10

tan θ = 0.7

θ = tan⁻¹(0.7)

θ ≈ 35.0°

θ ≈ 35.0°

---

Example 6: Finding an Angle Using Sine

Suppose:

Opposite = 9 m

Hypotenuse = 15 m

sin θ = 9 ÷ 15

sin θ = 0.6

θ = sin⁻¹(0.6)

θ ≈ 36.9°

θ ≈ 36.9°

---

Example 7: Finding an Angle Using Cosine

Suppose:

Adjacent = 12 cm

Hypotenuse = 20 cm

cos θ = 12 ÷ 20

cos θ = 0.6

θ = cos⁻¹(0.6)

θ ≈ 53.1°

θ ≈ 53.1°

---

Finding the Remaining Angle

The angles inside every triangle add to:

180°

In a right triangle, one angle is already 90°.

Therefore, the two remaining angles must add to:

90°

If one acute angle is θ and the other is φ:

θ + φ = 90°

So:

φ = 90° − θ

---

Example 8: Finding the Remaining Angle

Suppose one acute angle is:

37°

Then the other is:

90° − 37°

= 53°

The remaining angle is 53°.

This is usually faster than using another trigonometric calculation.

---

Complementary Angles

The two acute angles of a right triangle are complementary.

That means their sum is:

90°

For example:

20° + 70° = 90°

35° + 55° = 90°

48° + 42° = 90°

This provides an important way to check your answers.

---

Solving an Entire Triangle

Consider a right triangle with:

One acute angle = 32°

Hypotenuse = 15 cm

We want to determine:

• opposite side
• adjacent side
• remaining angle

This requires several steps.

---

Step 1: Find the Opposite Side

sin 32° = opposite ÷ 15

Opposite = 15 sin 32°

Opposite ≈ 7.95 cm

---

Step 2: Find the Adjacent Side

cos 32° = adjacent ÷ 15

Adjacent = 15 cos 32°

Adjacent ≈ 12.72 cm

---

Step 3: Find the Remaining Angle

Remaining angle:

90° − 32°

= 58°

Therefore the complete triangle contains:

Angles: 90°, 32°, 58°

Sides: 15 cm, 12.72 cm, 7.95 cm

The triangle is now solved.

---

There Is Often More Than One Method

Suppose you know:

Hypotenuse = 15 cm

Opposite = 9 cm

You could find the remaining shorter side using the Pythagorean theorem.

Or you could:

1. calculate an angle using inverse sine
2. use cosine or tangent to find the missing side

Both approaches can work.

Usually, choose the method that:

• uses the fewest steps
• uses the original information
• avoids unnecessary rounding

---

Example 9: Solving from Two Sides

Suppose:

Opposite = 9 cm

Adjacent = 12 cm

Find all remaining measurements.

First find the hypotenuse:

c² = 9² + 12²

c² = 81 + 144

c² = 225

c = 15 cm

Now find one angle:

tan θ = 9 ÷ 12

θ = tan⁻¹(9 ÷ 12)

θ ≈ 36.9°

Find the other angle:

90° − 36.9°

= 53.1°

Complete solution:

Sides: 9 cm, 12 cm, 15 cm

Angles: 36.9°, 53.1°, 90°

---

Example 10: Solving from a Side and an Angle

A right triangle has:

Adjacent side = 20 m

Angle = 41°

Find the opposite side, hypotenuse, and remaining angle.

Opposite Side

tan 41° = opposite ÷ 20

Opposite = 20 tan 41°

Opposite ≈ 17.39 m

Hypotenuse

cos 41° = 20 ÷ hypotenuse

Hypotenuse = 20 ÷ cos 41°

Hypotenuse ≈ 26.50 m

Remaining Angle

90° − 41° = 49°

Complete solution:

Adjacent = 20 m

Opposite ≈ 17.39 m

Hypotenuse ≈ 26.50 m

Angles = 41°, 49°, 90°

---

Multi-Step Triangle Problems

Some problems require information from one calculation before the next calculation can be completed.

For example:

1. Find a missing side using trigonometry.
2. Use that side in a second triangle.
3. Find another distance.
4. Interpret the final result.

Breaking a complicated diagram into smaller right triangles can make the problem much easier.

---

Example 11: Two Connected Triangles

A support structure contains two right triangles.

The first triangle has:

Angle = 30°

Adjacent side = 8 m

First find its vertical height:

tan 30° = h ÷ 8

h = 8 tan 30°

h ≈ 4.62 m

That height may then become a known side in another triangle.

If the second triangle has:

Opposite = 4.62 m

Angle = 40°

then:

tan 40° = 4.62 ÷ x

x = 4.62 ÷ tan 40°

x ≈ 5.51 m

The result from the first triangle became information needed for the second.

---

Draw Before You Calculate

For complicated problems, start with a diagram.

A useful diagram should show:

• right angles
• known angles
• known side lengths
• unknown quantities
• labels for important points

Do not worry if the diagram is not perfectly to scale.

Its purpose is to organize the mathematical information.

---

Mark the Reference Angle

Before labeling:

opposite

and:

adjacent

identify the angle you are working with.

The same side can be opposite one acute angle and adjacent to the other.

The hypotenuse, however, is always the same side.

---

A Decision Strategy

When solving a right triangle, ask:

Do I know two sides?

If yes, consider:

• Pythagorean theorem for the third side
• inverse trig for an angle

Do I know one side and one acute angle?

If yes, use:

• sine
• cosine
• tangent

depending on the sides involved.

Do I already know one acute angle?

Find the other using:

90° − known angle

This simple decision process can reduce unnecessary calculations.

---

Use Original Information When Possible

Suppose you calculate:

x ≈ 7.3 cm

and then use 7.3 cm in another calculation.

The actual calculator value may have been:

7.347829...

Using the rounded value introduces additional rounding error.

Whenever possible:

• use original measurements
• keep full calculator values during working
• round only the final answers

---

Verifying Your Solution

A complete triangle solution should be checked.

Several methods are available.

---

Check 1: Angle Sum

Your angles should add to:

180°

For a right triangle:

90° + angle A + angle B = 180°

Therefore:

angle A + angle B = 90°

If you obtain:

90°, 42°, and 58°

then:

90 + 42 + 58 = 190°

Something is wrong.

---

Check 2: Hypotenuse Must Be Longest

Suppose your calculated sides are:

6 cm

8 cm

7 cm

and you have identified 7 cm as the hypotenuse.

This cannot be correct.

The hypotenuse must be the longest side.

---

Check 3: Pythagorean Theorem

Suppose your calculated sides are approximately:

5 cm

12 cm

13 cm

Check:

5² + 12²

= 25 + 144

= 169

13² = 169

The sides satisfy the Pythagorean theorem.

This supports the solution.

---

Check 4: Use a Different Trigonometric Ratio

Suppose:

Opposite = 6 cm

Adjacent = 8 cm

Hypotenuse = 10 cm

For angle θ:

sin θ = 6/10

and:

tan θ = 6/8

Both calculations should produce the same angle, allowing for rounding.

This provides an independent check.

---

Check 5: Compare Angles and Sides

Larger angles should be opposite longer sides.

Suppose one angle is:

65°

and another is:

25°

The side opposite 65° should be longer than the side opposite 25°.

If your answer contradicts this relationship, check your work.

---

Check 6: Estimate Before Calculating

Suppose:

Angle = 45°

Adjacent side = 10 cm

At 45°, the two shorter sides of a right triangle are equal.

Therefore, the opposite side should be approximately:

10 cm

If your calculator gives:

1.0 cm

or:

100 cm

you should immediately suspect an error.

---

Special Right Triangle: 45°–45°–90°

A right triangle with one angle of 45° must have:

45°, 45°, 90°

The two shorter sides are equal.

If each shorter side is:

x

then the hypotenuse is:

x√2

For example:

Short sides = 5 cm and 5 cm

Hypotenuse:

5√2 ≈ 7.07 cm

---

Special Right Triangle: 30°–60°–90°

Another useful right triangle has angles:

30°, 60°, 90°

Its side ratio is:

1 : √3 : 2

where:

• shortest side is opposite 30°
• longer leg is opposite 60°
• hypotenuse is opposite 90°

Recognizing these patterns can sometimes make calculations faster.

---

Communicating a Complete Solution

A strong mathematical solution should show enough information that another person can follow your reasoning.

For example:

Given:

θ = 38°

Adjacent = 12.0 m

Find: Opposite side.

Method:

tan θ = opposite ÷ adjacent

tan 38° = x ÷ 12.0

x = 12.0 tan 38°

x ≈ 9.38 m

Answer:

The opposite side is approximately 9.38 m.

This is clearer than writing only:

9.38

---

Include Units

If the sides represent lengths, include appropriate units.

Examples:

8.4 cm

12.7 m

3.2 km

Angles should include:

°

For example:

θ = 37.5°

Units help communicate what the answer represents.

---

Use Appropriate Precision

Suppose the measurements are given as:

12.0 m

and:

35°

It would usually be unreasonable to report:

8.402574931 m

A more appropriate answer might be:

8.4 m

or:

8.40 m

depending on the context and expected precision.

---

Worked Example: Complete Triangle

A right triangle has:

Hypotenuse = 18 cm

Angle A = 37°

Find:

• side opposite A
• side adjacent to A
• angle B

Opposite Side

sin 37° = opposite ÷ 18

Opposite = 18 sin 37°

Opposite ≈ 10.83 cm

Adjacent Side

cos 37° = adjacent ÷ 18

Adjacent = 18 cos 37°

Adjacent ≈ 14.38 cm

Remaining Angle

B = 90° − 37°

B = 53°

Complete solution:

Sides ≈ 10.83 cm, 14.38 cm, 18 cm

Angles = 37°, 53°, 90°

---

Worked Example: Two Known Legs

A right triangle has shorter sides:

16 m

and:

30 m

Find all remaining information.

Hypotenuse

c² = 16² + 30²

c² = 256 + 900

c² = 1156

c = 34 m

First Angle

Let θ be opposite the 16 m side.

tan θ = 16 ÷ 30

θ = tan⁻¹(16/30)

θ ≈ 28.1°

Second Angle

90° − 28.1°

= 61.9°

Complete solution:

Sides = 16 m, 30 m, 34 m

Angles ≈ 28.1°, 61.9°, 90°

---

Worked Example: Hypotenuse and One Leg

Suppose:

Hypotenuse = 25 cm

One leg = 15 cm

Find the missing side and angles.

Missing Side

15² + b² = 25²

225 + b² = 625

b² = 400

b = 20 cm

Angle Opposite the 15 cm Side

sin θ = 15 ÷ 25

θ = sin⁻¹(0.6)

θ ≈ 36.9°

Remaining Angle

90° − 36.9°

= 53.1°

Complete solution:

Sides = 15 cm, 20 cm, 25 cm

Angles ≈ 36.9°, 53.1°, 90°

---

Worked Example: One Side and One Angle

Suppose:

Angle A = 63°

Adjacent side = 7.5 m

Find the opposite side, hypotenuse, and remaining angle.

Opposite

tan 63° = opposite ÷ 7.5

Opposite = 7.5 tan 63°

Opposite ≈ 14.72 m

Hypotenuse

cos 63° = 7.5 ÷ hypotenuse

Hypotenuse = 7.5 ÷ cos 63°

Hypotenuse ≈ 16.52 m

Remaining Angle

90° − 63°

= 27°

Complete solution:

Sides ≈ 7.5 m, 14.72 m, 16.52 m

Angles = 27°, 63°, 90°

---

Multi-Step Example: Support Cable

A vertical pole is supported by a cable.

The cable is:

18 m long

and makes an angle of:

55°

with the ground.

Find:

1. the height where the cable attaches to the pole
2. the horizontal distance from the pole to the cable anchor

Height

sin 55° = height ÷ 18

height = 18 sin 55°

height ≈ 14.74 m

Horizontal Distance

cos 55° = distance ÷ 18

distance = 18 cos 55°

distance ≈ 10.32 m

The cable attaches approximately 14.74 m above the ground and is anchored approximately 10.32 m horizontally from the pole.

---

Multi-Step Example: Roof

One half of a symmetrical roof forms a right triangle.

Horizontal run = 5.5 m

Roof angle = 32°

Find:

1. the vertical rise
2. the sloping roof length

Rise

tan 32° = rise ÷ 5.5

rise = 5.5 tan 32°

rise ≈ 3.44 m

Roof Length

cos 32° = 5.5 ÷ roof length

Roof length = 5.5 ÷ cos 32°

roof length ≈ 6.49 m

The roof rises approximately 3.44 m and has a sloping length of approximately 6.49 m.

---

Multi-Step Example: Ladder

A 7 m ladder leans against a wall.

Its base is:

2.5 m

from the wall.

Find:

1. the height reached
2. the angle the ladder makes with the ground

Height

h² + 2.5² = 7²

h² = 49 − 6.25

h² = 42.75

h ≈ 6.54 m

Angle

cos θ = 2.5 ÷ 7

θ = cos⁻¹(2.5/7)

θ ≈ 69.1°

The ladder reaches approximately 6.54 m up the wall and makes an angle of approximately 69.1° with the ground.

---

Multi-Step Problems with Shared Sides

Some diagrams contain several right triangles that share a side.

A useful strategy is:

Triangle 1 → calculate shared side → Triangle 2 → calculate final unknown

Do not try to solve the entire diagram in one step.

Identify one triangle containing enough information to begin.

Then use its result in the next triangle.

---

Choosing the Most Efficient Method

Suppose two sides are known.

You could sometimes use several methods.

For example:

Legs = 8 cm and 15 cm

To find the hypotenuse, the most direct method is:

Pythagorean theorem

rather than finding an angle first and then using sine or cosine.

Efficient mathematical reasoning means choosing a method that minimizes unnecessary steps.

---

Avoiding Unnecessary Calculations

Suppose one acute angle is:

24°

You do not need trigonometry to find the other acute angle.

Simply calculate:

90° − 24°

= 66°

Use the simplest valid method.

---

Common Mistake: Using the Wrong Angle

Suppose the side you call "opposite" is actually opposite the other acute angle.

Your trig equation will be incorrect.

Always mark the angle you are working from before labeling O and A.

---

Common Mistake: Treating the Hypotenuse as Adjacent

The hypotenuse touches both acute angles, but it is never called the adjacent side in SOH–CAH–TOA.

The adjacent side means the shorter side beside the reference angle.

---

Common Mistake: Forgetting Inverse Trig

If you know:

tan θ = 0.75

you cannot simply write:

θ = 0.75°

Instead:

θ = tan⁻¹(0.75)

θ ≈ 36.9°

---

Common Mistake: Using Inverse Trig to Find a Side

Inverse trig functions are primarily used to determine angles.

If you know:

θ = 40°

Adjacent = 12 m

and need the opposite side, use:

tan 40° = opposite ÷ 12

not inverse tangent.

---

Common Mistake: Calculator Mode

If angles are given in degrees, your calculator should normally be in:

DEG

not:

RAD

A calculator in the wrong mode can produce completely different answers.

---

Common Mistake: Rounding Too Early

Suppose an intermediate result is:

13.728419...

Do not immediately round it to:

14

if it will be used in another calculation.

Keep the calculator value and round the final answer.

This reduces accumulated rounding error.

---

Common Mistake: Incomplete Answers

If the question says:

Solve the triangle

finding only one missing side is not enough.

You should determine:

• all three sides
• all three angles

and present them clearly.

---

A Reliable Solving Strategy

When asked to solve a right triangle:

Step 1: Draw or inspect the triangle.

Step 2: Identify the 90° angle.

Step 3: Identify the hypotenuse.

Step 4: Record all known sides and angles.

Step 5: Decide which unknown should be found first.

Step 6: Select Pythagoras, sine, cosine, tangent, or inverse trig.

Step 7: Write the equation.

Step 8: Substitute known values.

Step 9: Solve without excessive intermediate rounding.

Step 10: Find the remaining measurements.

Step 11: Verify the solution.

Step 12: Present all sides, angles, units, and appropriate rounding.

---

A Triangle-Solving Decision Guide

Two sides known?

→ Use Pythagorean theorem for the third side.

→ Use inverse trig for an angle.

→ Use 90° minus that angle for the final angle.

One side + one acute angle known?

→ Use SOH–CAH–TOA to calculate missing sides.

→ Use 90° minus the known acute angle for the remaining angle.

Two acute angles known?

→ Their values alone determine the shape but not the size of the triangle.

At least one side length is required to determine actual side lengths.

---

Verifying a Complete Solution

Suppose you obtain:

Sides:

8.0 cm, 15.0 cm, 17.0 cm

Angles:

28.1°, 61.9°, 90°

Check the angles:

28.1 + 61.9 + 90 = 180°

Check the sides:

8² + 15² = 64 + 225

= 289

17² = 289

The solution passes both checks.

---

Communicating Mathematical Reasoning

A complete solution should normally contain:

Diagram or labels

Show what each value represents.

Formula or ratio

For example:

tan θ = O/A

Substitution

tan θ = 8/15

Calculation

θ = tan⁻¹(8/15)

θ ≈ 28.1°

Final statement

The angle is approximately 28.1°.

This allows another person to understand and verify your reasoning.

---

Real-World Interpretation

Right triangles appear in many situations involving:

• ramps
• roofs
• ladders
• roads
• bridges
• navigation
• surveying
• support cables
• heights
• distances
• slopes

In real-world problems, the final answer should describe what the calculated value actually represents.

Instead of:

x = 12.6

write:

The support cable is approximately 12.6 m long.

---

Did You Know?

Right triangle trigonometry provides several independent ways to check a solution.

If you know all three sides, you can:

• check them with the Pythagorean theorem
• calculate an angle using sine
• calculate the same angle using cosine
• calculate it again using tangent

All of these methods should produce consistent results, allowing for rounding.

This is an important mathematical habit:

Do not only calculate an answer — look for evidence that the answer makes sense.

---

Key Terms

• Right triangle: Triangle containing one 90° angle.
• Hypotenuse: Longest side of a right triangle, opposite the right angle.
• Opposite side: Side across from the reference angle.
• Adjacent side: Non-hypotenuse side beside the reference angle.
• Sine: Ratio of opposite side to hypotenuse.
• Cosine: Ratio of adjacent side to hypotenuse.
• Tangent: Ratio of opposite side to adjacent side.
• Inverse sine: Function used to determine an angle from a sine ratio.
• Inverse cosine: Function used to determine an angle from a cosine ratio.
• Inverse tangent: Function used to determine an angle from a tangent ratio.
• Pythagorean theorem: Relationship between the three sides of a right triangle.
• Complementary angles: Two angles whose measures total 90°.
• Reference angle: Acute angle used when identifying opposite and adjacent sides.
• Solve a triangle: Determine all unknown sides and angles.
• Multi-step problem: Problem requiring more than one mathematical calculation or relationship.
• Verification: Process of checking whether a solution is mathematically reasonable and consistent.

---

Key Relationships

For right triangles:

a² + b² = c²

where c is the hypotenuse.

Trigonometric ratios:

sin θ = opposite ÷ hypotenuse

cos θ = adjacent ÷ hypotenuse

tan θ = opposite ÷ adjacent

Inverse trigonometry:

θ = sin⁻¹(O/H)

θ = cos⁻¹(A/H)

θ = tan⁻¹(O/A)

Acute-angle relationship:

Angle A + Angle B = 90°

Whole triangle:

Angle A + Angle B + 90° = 180°

---

Key Takeaways

• Solving a right triangle means determining all unknown sides and angles.
• A right triangle already contains one 90° angle.
• The two acute angles of a right triangle are complementary.
• The Pythagorean theorem can find a missing side when two sides are known.
• Sine relates the opposite side and hypotenuse.
• Cosine relates the adjacent side and hypotenuse.
• Tangent relates the opposite and adjacent sides.
• Inverse trigonometric functions are used to calculate unknown angles from side ratios.
• SOH–CAH–TOA helps identify the appropriate trigonometric ratio.
• Opposite and adjacent sides depend on the chosen reference angle.
• The hypotenuse is always opposite the right angle and is always the longest side.
• When two sides are known, Pythagoras and inverse trigonometry can usually solve the triangle.
• When one side and one acute angle are known, trigonometric ratios can usually solve the triangle.
• Knowing only the angles determines the shape of a triangle but not its actual size.
• Multi-step problems should be divided into smaller calculations.
• In compound diagrams, one calculated side may become a known side in another triangle.
• Original information should be used whenever possible to reduce rounding error.
• Intermediate values should not be rounded excessively.
• Solutions can be checked using angle sums, the Pythagorean theorem, alternative trig ratios, and estimation.
• The two acute angles should always add to 90°.
• The hypotenuse must always be the longest side.
• Larger angles should be opposite longer sides.
• Efficient problem solving means selecting the simplest appropriate method.
• Complete mathematical solutions should show the method, substitution, calculation, units, and final answer.
• A numerical result should be interpreted in the context of the problem.
• Good mathematics includes both solving and verifying the solution.