Solve real-world acute-triangle problems

Ontario Grade 11 Mathematics

Trigonometric Functions

Represent the situation, choose a law, and check the result

A triangle can model distances between places or points that are hard to measure directly. An acute triangle has three angles, each less than

90^\circ

. To solve a real-world problem, turn the story into a labelled triangle. Then choose a rule that connects what is known to what is unknown. This lesson reviews useful triangle facts, explains the sine law and cosine law, and uses them to find a distance.

What you will learn

Recognize the sides and angles in an acute triangle.
Choose the sine law or cosine law to match the measurements given.
Solve a real-world triangle problem using a labelled sketch, an equation, and appropriate units.
Check that an answer is reasonable and state it in context.

1. Prerequisite bridge: read and label the triangle

A triangle has three sides and three interior angles. An acute angle is greater than

0^\circ

and less than

90^\circ

. The interior angles of every triangle add to

180^\circ

. These facts help you label a diagram and check a result.

Use capital letters for the angles and the matching lowercase letters for the opposite sides. For example, side

a

is across from angle

A

. The word opposite means across from, not next to. Matching each side with its opposite angle is important when using the sine law.

A sketch does not need to be drawn to scale. Its job is to show how the measurements fit together. Label the measurements from the story, and mark the unknown with a variable. If the measurements use different units, convert them to one unit before calculating. For example, change kilometres to metres if the other distances are in metres.

When an angle is given in degrees, make sure your calculator is set to degree mode. Follow the calculator instructions for entering the angle and the trigonometric function.

A+B+C=180^\circ

The side and angle with the same letter are opposite each other.
The three interior angles add to $180^\circ$ .
Use consistent units, and use degree mode for angles given in degrees.

2. Choose the law that fits the information

The sine law relates the length of a side to the sine of its opposite angle. The sine of an angle is a value found using a calculator. Look for a known side and its opposite angle. That matching pair can help you find another side or angle.

The cosine law relates three sides to one angle. It is useful when you know two sides and the angle between them. That angle is called the included angle. The cosine law can also find an angle when all three side lengths are known.

Before choosing a law, list the known measurements and identify what is unknown. Then check how the measurements are arranged in your triangle. Having an angle in the problem does not automatically mean the sine law is the right choice. If the angle is between two known sides, the cosine law is a direct fit.

Use enough calculator digits during the calculation. Round only the final result, unless the question asks for a different level of accuracy. A final measurement should have units and should answer the question in the story.

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Use the sine law when a known opposite side-angle pair is available.
Use the cosine law for two sides and their included angle, or for three known sides.
Choose a law by looking at how the known measurements fit together.

3. Represent the situation and check the answer

A clear solution has three connected parts: a labelled sketch, an equation, and a sentence that interprets the answer. The sketch translates the story into a triangle. The equation shows how the known measurements are related. The sentence tells what the calculated value means and gives its units.

For example, two distances from a landmark to two survey points form two sides of a triangle. If the angle between those distances is known, the unknown distance between the survey points is opposite that angle. This is a two-sides-and-included-angle situation, so use the cosine law.

Once you calculate, check whether the result fits the situation. A side length must be positive. The angles in an acute triangle must each be less than

90^\circ

and must add to

180^\circ

. Compare the size of the answer with the given distances. A result that is much too large or too small may signal a setup or calculator error.

Real measurements are often approximate. Your answer should not suggest more precision than the measurements support. If the question asks for the nearest metre, keep calculator values during the work and round the final distance to the nearest metre.

a^2=b^2+c^2-2bc\cos A

Translate the story into sides and angles before substituting values.
Check that the result is positive and fits the scale of the situation.
Give a rounded answer with units and explain what it represents.

4. Apply the process to a surveying situation

A surveyor may be unable to measure the distance between two points directly. Instead, the surveyor can measure from each point to a landmark and measure the angle where the two sight lines meet. Those three pieces of information form a triangle. The cosine law finds the distance between the two points.

For a different arrangement, a known side and its opposite angle may be available. In that case, the sine law can connect that pair to another side-angle pair. The diagram helps you decide which arrangement you have; the law is not chosen just because it appears familiar.

A final check is part of solving the problem, not an extra calculation. Confirm that the equation matches the labelled triangle, that the calculator used degrees, and that the answer makes sense in the real setting. Then report the requested quantity in a complete sentence.

c^2=a^2+b^2-2ab\cos C

A real-world triangle is a model of the situation, not necessarily a scale drawing.
Match the law to the known information before calculating.
Use the final sentence to connect the number and its units to the problem.

Worked example

Find the distance between two survey points

A surveyor stands at two points,

P

and

Q

. A landmark is at

R

. The distances

PR

and

QR

are

120

m and

95

m. The angle between the two sight lines at

R

is

68^\circ

. Find the distance

PQ

to the nearest metre.

Sketch and identify
Draw triangle $PQR$ . The two known sides meet at $R$ , and the unknown distance $PQ$ is opposite the known angle there. This is a two-sides-and-included-angle situation.
$PR=120\text{ m},\quad QR=95\text{ m},\quad \angle R=68^\circ$
Choose the cosine law
The cosine law connects the two known sides and the angle between them to the opposite side. Let $PQ=x$ .
$x^2=120^2+95^2-2(120)(95)\cos 68^\circ$
Calculate
Evaluate the expression in degree mode. Take the positive square root because a distance cannot be negative. The result is approximately $122.0$ m before rounding.
$x\approx122.0\text{ m}$
Interpret and check
The distance is positive and is less than the sum of the two distances through the landmark. To the nearest metre, the distance between the survey points is $122$ m.
$PQ\approx122\text{ m}$

Answer: The distance between

P

and

Q

is approximately

122

m.

Check: With the same two side lengths, an angle of

68^\circ

gives a shorter opposite side than a right angle would. The calculated distance is also less than

120+95=215

m, so it is reasonable.

Common mistakes and how to avoid them

Using the sine law just because the problem gives an angle.

Correction: Check whether the angle is between two known sides. If it is, use the cosine law. Use the sine law when a known side and its opposite angle form a pair.

Matching a side with an angle that is beside it rather than opposite it.

Correction: Label the triangle so each lowercase side letter matches the capital letter of the angle across from it.

Using a calculator setting that does not match the angle units.

Correction: When the problem gives degrees, set the calculator to degree mode before evaluating a trigonometric expression.

Rounding during the calculation or leaving off units.

Correction: Keep calculator values until the final step. Round as requested and include the correct units in the answer.

Lesson summary

Draw and label a triangle that represents the situation.
Use the sine law when a known opposite side-angle pair is available.
Use the cosine law for two sides and their included angle, or for three known sides.
Check the calculation and answer in context, then round suitably and include units.

Check your understanding

Question 1

Two sides of an acute triangle are

7

cm and

10

cm. The angle between them is

42^\circ

. Which method directly finds the third side?

The sine law
The cosine law
Add the two side lengths
Subtract the shorter side length from the longer side length

Show answer and explanation

The cosine law

The known angle is between the two known sides. The cosine law relates those measurements to the third side.

Question 2

In a labelled triangle, side

c

is opposite which angle?

Angle $A$
Angle $B$
Angle $C$
The largest angle only

Show answer and explanation

Angle

C

The lowercase side letter matches the capital letter of its opposite angle, so side

c

is opposite angle

C

.

Question 3

A triangle has angles

48^\circ

and

67^\circ

. What is its third angle?

$55^\circ$
$65^\circ$
$75^\circ$
$115^\circ$

Show answer and explanation

65^\circ

Triangle angles add to

180^\circ

. The third angle is

180^\circ-48^\circ-67^\circ=65^\circ

.

Key terms

Acute angle: An angle greater than $0^\circ$ and less than $90^\circ$ .
Opposite side: The side across from a given angle.
Included angle: The angle between two specified sides.
Sine law: A rule that relates each side of a triangle to the sine of its opposite angle.
Cosine law: A rule that relates the sides of a triangle to the cosine of one of its angles.

Continue through MCF3M

View the complete MCF3M Ontario Grade 11 Mathematics curriculum and lessons

Tutor support for C1.5 · Solve real-world acute-triangle problems

Working on C1.5 · Solve real-world acute-triangle problems in MCF3M? Bring the question and your own attempt. In a live session, you can show each step of your attempt and compare it with a worked example. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث C1.5 · حل مسائل واقعی مثلث حاده در درس MCF3M، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید گام‌های تلاش خود را نشان دهید و آن را با یک نمونهٔ حل‌شده مقایسه کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C1.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question