T10 Solve real problems involving acute triangles

Ontario Grade 10 Mathematics

Trigonometry

Choose a triangle rule, match sides and angles, and interpret the result

You already know that a triangle’s angles add to

180^\circ

. You may also have used right-triangle ratios, but those ratios need a right angle. In this lesson, you will solve problems involving acute triangles. An acute triangle has three angles smaller than

90^\circ

. The sine law and cosine law let you find missing measurements when the triangle does not have a right angle. You will learn how to identify the information you have, choose a rule, and check your answer in context.

What you will learn

Recognize an acute triangle and identify opposite side-and-angle pairs.
Choose the sine law or cosine law to match the information in a problem.
Solve a real-world problem involving an acute triangle and check that the answer is reasonable.

1. Review triangle parts and angle facts

A triangle has three sides and three angles. The side opposite an angle is the side directly across from it. For example, the side across from angle

A

is called side

a

. Side

b

is across from angle

B

, and side

c

is across from angle

C

. These matching pairs are important when you choose a rule.

An acute triangle has three angles, each smaller than

90^\circ

. The angle-sum rule still applies: the three angles add to

180^\circ

. If two angles are known, subtract their sum from

180^\circ

to find the third. This can help complete a sketch or check a result.

In a real problem, measurements may describe roads, paths, supports, or distances between locations. First draw a simple triangle to represent the situation. The sketch does not need to be to scale. Label the known measurements and mark the measurement you need to find.

A+B+C=180^\circ

Match each side with the angle directly across from it.
An acute triangle has three angles below $90^\circ$ .
Use the angle-sum rule to find a missing angle when the other two are known.

2. Choose a rule that fits the known information

The sine law connects a triangle’s sides with the sines of their opposite angles. The sine of an angle is a value found using a calculator. Use the sine law when you know at least one opposite side-and-angle pair and need another side or angle. A matching pair means that the side and angle are directly across from each other.

The cosine law connects two sides and the angle between them to the third side. The angle between two sides is called the included angle. Use the cosine law when you know two sides and their included angle. It can also be rearranged to find an angle when all three sides are known.

A quick way to choose is to circle the known measurements on your sketch. Look for an opposite side-and-angle pair. If you have one, the sine law may fit. If you have two sides and the angle between them, the cosine law fits. Write the labels before substituting numbers so you do not mix up an angle and a side.

Set your calculator to degree mode. The angle measurements in these problems are given in degrees. Keep calculator values through the calculation and round only when you reach the requested final answer.

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

The sine law uses opposite side-and-angle pairs.
The cosine law uses two sides and their included angle to find the third side.
The cosine law can also find an angle when all three sides are known.
Check that your calculator is in degree mode.

3. Read the cosine law from a sketch

Imagine two straight paths that begin at the same point. One path is

42

m long and the other is

57

m long. The angle where they meet is

68^\circ

. The straight-line distance between the far ends completes a triangle. On a sketch, the unknown distance lies across from the known angle.

For this arrangement, call the unknown side

c

, the two known sides

a

and

b

, and the angle between the known sides

C

. The cosine law subtracts a term involving the two side lengths and the cosine of the included angle. This subtraction is part of the rule; do not replace it with addition.

The sketch gives a useful reasonableness check. The angle is acute, so the distance across the ends should be less than it would be if the paths opened to a right angle. It should also be longer than the difference between the two path lengths. A calculated value outside a reasonable range may signal a setup or calculator error.

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

The side found with the cosine law is opposite the included angle.
Use the angle formed by the two known sides.
Estimate the size of an answer before calculating.

4. Plan, solve, and practise

A reliable plan has four parts. Draw and label the triangle. Choose a rule that uses the known information. Substitute values with their units and calculate. Then state the result with units and check whether it makes sense in the situation.

For a sine-law problem, check that every side is paired with its opposite angle. For a cosine-law problem with two sides, check that the angle is between those sides. These checks matter more than the order in which you write the labels.

Independent practice: A surveyor marks two straight distances from one point, one measuring

35

m and the other

48

m. The angle between them is

72^\circ

. Sketch the triangle, identify the rule that fits, and write an expression for the distance between the two marked endpoints. Do not round an intermediate value.

Next, consider a triangular sign support. One side is

6

m, another is

9

m, and the angle between them is

55^\circ

. Decide which side is opposite the

55^\circ

angle and describe what you would calculate first. In both situations, your sketch should show why the chosen rule matches the measurements.

Sketch first, then choose the rule from the measurements shown.
Keep units attached to real measurements.
Round at the end and check the answer against the sketch.

Choose a triangle rule

Known information	Useful rule	What to check
An opposite side-and-angle pair, plus another side or angle	Sine law	Match each side with its opposite angle
Two sides and the angle between them	Cosine law	Use the angle formed by the two known sides
All three sides and a missing angle	Cosine law	Keep track of which side is opposite the angle you need

Worked example

Find the distance across a triangular park

Two paths from a meeting point to the ends of a park are

42

m and

57

m long. The angle between the paths is

68^\circ

. Find the straight-line distance between the two ends of the park, to the nearest tenth of a metre.

Sketch and choose
Draw the two known paths meeting at an angle of $68^\circ$ . The distance between their far ends is opposite that angle. Since the problem gives two sides and their included angle, the cosine law fits.
Substitute
Let $c$ be the distance to find. Use the two path lengths for $a$ and $b$ , and use the angle between them for $C$ .
$c^2=42^2+57^2-2(42)(57)\cos 68^\circ$
Calculate and round
The squared distance is about $3219.38$ . Take the positive square root because a distance is positive, then round to the nearest tenth.
$c\approx\sqrt{3219.38}\approx56.7$

Answer: The straight-line distance between the ends of the park is about

56.7

m.

Check: The two paths are

42

m and

57

m long. The answer is less than the longer path, as expected for this acute included angle, and it is greater than the difference between the path lengths, which is

15

m. The units are metres, matching the measurements in the problem.

Common mistakes and how to avoid them

Using the sine law without matching each side to its opposite angle.

Correction: Label the triangle first. Side

a

matches angle

A

, side

b

matches angle

B

, and side

c

matches angle

C

.

Using an angle that is not between the two known sides in the cosine law.

Correction: On your sketch, identify the angle formed by the two sides you know. That is the included angle for this use of the cosine law.

Changing the subtraction in the cosine law to addition.

Correction: Copy the cosine-law pattern carefully. The term involving the two known sides and the cosine is subtracted.

Rounding intermediate values heavily.

Correction: Keep the calculator value until the final step, then round as the question requests.

Giving a distance without units.

Correction: Include the unit used in the problem, such as metres or centimetres.

Lesson summary

An acute triangle has three angles smaller than $90^\circ$ .
The angles in every triangle add to $180^\circ$ .
Use the sine law when an opposite side-and-angle pair is available.
Use the cosine law for two sides and their included angle, or for three known sides when finding an angle.
Draw and label a sketch, check the answer, and include units.

Check your understanding

Question 1

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

The angles total

180^\circ

. Subtract the two known angles:

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

.

Question 2

You know two sides of an acute triangle and the angle between them. Which rule should you use to find the third side?

The sine law
The cosine law
The angle-sum rule alone
A right-triangle ratio

Show answer and explanation

The cosine law

The cosine law uses two sides and their included angle to find the side opposite that angle.

Question 3

In triangle

ABC

, side

a

is opposite which angle?

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

Show answer and explanation

Angle

A

The standard labels in this lesson pair side

a

with the angle directly across from it, angle

A

.

Question 4

Two sides are

8

cm and

11

cm, and their included angle is

60^\circ

. Which expression gives the square of the side opposite the

60^\circ

angle?

$8^2+11^2-2(8)(11)\cos 60^\circ$
$8^2+11^2+2(8)(11)\cos 60^\circ$
$\frac{8}{\sin 60^\circ}=\frac{11}{\sin 60^\circ}$
$(8+11)\cos 60^\circ$

Show answer and explanation

8^2+11^2-2(8)(11)\cos 60^\circ

The cosine law subtracts twice the product of the known sides and the cosine of the included angle.

Key terms

Acute triangle: A triangle with three angles smaller than $90^\circ$ .
Opposite side: The side directly across from a particular angle in a triangle.
Included angle: The angle formed by 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 two sides and their included angle to the third side, and can be rearranged to find an angle.

Continue through MPM2D

View the complete MPM2D Ontario Grade 10 Mathematics curriculum and lessons

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T10. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

T10 · Solve real problems involving acute triangles

Ontario Grade 10 Mathematics

Choose a triangle rule, match sides and angles, and interpret the result

What you will learn

1. Review triangle parts and angle facts

2. Choose a rule that fits the known information

3. Read the cosine law from a sketch

4. Plan, solve, and practise

Choose a triangle rule

Find the distance across a triangular park

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MPM2D

About this lesson