Choose and apply the sine law or cosine law in acute triangles

Ontario Grade 11 Mathematics

Trigonometric Functions

How to match the information in an acute triangle to the right rule

A map shows two paths that meet at an angle. You know the lengths of the paths and the angle where they meet, but not the straight-line distance between their endpoints. The cosine law can help find that distance. If you know a side and its opposite angle, along with another angle, the sine law may help instead. The important first step is to label the triangle and match each piece of information to the rule that uses it.

What you will learn

Identify the side opposite each angle in a triangle.
Choose the sine law or cosine law from the information given.
Apply the selected law to find a missing side or angle in an acute triangle.
Check whether a calculated result is reasonable and use suitable units.

1. Prerequisite bridge: label the triangle

A triangle has three sides and three interior angles. The interior angles add to

180^\circ

. An acute angle measures less than

90^\circ

. An acute triangle has three acute angles.

In triangle

ABC

, capital letters name the angles. The matching lowercase letters name the opposite sides: side

a

is opposite angle

A

, side

b

is opposite angle

B

, and side

c

is opposite angle

C

. Opposite means directly across from an angle, not one of the sides that forms it.

A sketch helps you see which sides meet at an angle. It does not need to be drawn to scale. Mark every known side and angle, then mark the quantity you need to find. This small step helps prevent pairing the wrong values in a formula.

The sine of an angle is a trigonometric value that you can find with a calculator. The inverse sine and inverse cosine calculator functions can find an angle from a known sine or cosine value. When the problem gives angles in degrees, set the calculator to degree mode.

A+B+C=180^\circ

Match angle $A$ with opposite side $a$ , angle $B$ with side $b$ , and angle $C$ with side $c$ .
An acute angle is less than $90^\circ$ .
Label a sketch before choosing a law.

2. Choose the law that fits

The sine law connects a side length with the sine of its opposite angle. Use it when the known information includes at least one opposite side-angle pair and you need another side or angle. For example, if you know

a

,

A

, and

B

, you can use the sine law to find

b

. The known pair is

a

and

A

, because those values are opposite each other.

The cosine law connects three side lengths with an angle. Use it when you know two sides and the angle between them and need the third side. The angle between two sides is called the included angle. You can also use the cosine law when all three sides are known and you need an angle.

A practical choice process is to list what you know before calculating. Look first for an opposite side-angle pair. If you have one and need another matching pair, consider the sine law. If you have two sides and their included angle, or three sides, consider the cosine law. The known information must fit the law; seeing the unknown in a formula is not enough.

In the cosine law for finding side

c

, the angle is

C

, opposite the side being found. The other two sides are

a

and

b

. For a different target side, use the angle opposite that side and the other two side lengths.

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

The sine law uses opposite side-angle pairs.
The cosine law uses two sides and their included angle, or three sides.
Check that the angle used in the cosine law is opposite the side being found.

3. Read and use the formulas carefully

For the sine law, the numerator and denominator in each fraction belong together: a side is paired with the sine of its opposite angle. You usually only need to set two of the matching fractions equal. For example, to find

b

from

a

,

A

, and

B

, use the fraction for

a

with

A

and the fraction for

b

with

B

.

For the cosine law, keep track of the included angle. If sides

a

and

b

meet at angle

C

, then the opposite side is

c

. The subtraction term uses both side lengths and the cosine of that included angle.

To find an angle from three side lengths, rearrange the cosine law so that the cosine of the unknown angle is by itself. Then use inverse cosine on the calculator. A calculator result should be interpreted as an angle, not as a side length.

Keep calculator values unrounded until the last step. Round only as the question requests. Side lengths should have length units, such as centimetres or metres. Angles should be reported in the unit used by the question, usually degrees.

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

Do not match a side with the sine of a different, non-opposite angle.
Use the angle between the two known sides when applying the cosine law to find a third side.
Use inverse cosine when the cosine of an angle has been found.

4. Apply the choice and check the result

Consider a triangle with two known sides,

a=7.0\text{ cm}

and

b=9.0\text{ cm}

, and included angle

C=60^\circ

. The known angle is between the two known sides. That information points to the cosine law, not the sine law. The side opposite angle

C

is

c

, so the cosine law finds the required side directly.

After substituting, use the calculator in degree mode. The cosine of

60^\circ

is

0.5

. The resulting square of the side is

67

, so the side is the positive square root of

67

. Side lengths are positive, and the question asks for a length.

A quick reasonableness check can catch errors. With sides of

7.0

cm and

9.0

cm, the third side must be longer than their difference and shorter than their sum. The calculated value is between those limits. Also check that the unit is centimetres and that the answer has been rounded as requested.

In a practical setting, the same choice process applies to a map, a field, or a structure. Draw the triangle formed by the known distances and angle. Identify whether the given angle is between the known sides or opposite a known side. Then select the law that matches the labelled information.

The known side-angle arrangement determines the law to use.
A triangle side must be positive and have a length unit.
Check the result against the known side lengths and the requested rounding.

Which law matches the known information?

Known information	Law to consider	Why it fits
An opposite side-angle pair and another angle or side	Sine law	It connects each side to the sine of its opposite angle.
Two sides and the angle between them	Cosine law	It uses the included angle and the three side lengths.
All three side lengths and an angle to find	Cosine law	It can be rearranged to find the cosine of the angle.

Worked example

Find a third side with the cosine law

In an acute triangle,

a=7.0\text{ cm}

,

b=9.0\text{ cm}

, and included angle

C=60^\circ

. Find

c

to the nearest tenth of a centimetre.

Choose the law
Angle $C$ is between the two known sides, $a$ and $b$ . The cosine law uses two sides and their included angle to find the opposite side, $c$ .
Substitute the values
Put the known side lengths and angle into the cosine law for side $c$ . Use degree mode because the angle is given in degrees.
$c^2=7.0^2+9.0^2-2(7.0)(9.0)\cos 60^\circ$
Calculate and round
Since the cosine of $60^\circ$ is $0.5$ , simplify the right side. Take the positive square root because a side length is positive, then round to the nearest tenth.
$c^2=67,\qquad c=\sqrt{67}\approx 8.2\text{ cm}$

Answer:

c\approx 8.2\text{ cm}

Check: The difference of the known sides is

2.0

cm and their sum is

16.0

cm. The result is between those values, so it is a reasonable side length.

Common mistakes and how to avoid them

Pairing angle

A

with side

b

in the sine law.

Correction: Angle

A

is opposite side

a

. Pair each angle with its matching lowercase side.

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

Correction: Check which sides meet at the angle. The included angle is the one formed by the two known sides.

Using the wrong angle in the cosine law when finding a side.

Correction: For side

c

, use the angle opposite it,

C

, and the other sides,

a

and

b

.

Rounding an intermediate calculator value too early.

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

Lesson summary

Label each side-angle opposite pair before calculating.
Choose the sine law when an opposite side-angle pair connects to the unknown.
Choose the cosine law for two sides and their included angle, or for three known sides.
Check calculator mode, units, rounding, and whether the answer is reasonable.

Check your understanding

Question 1

You know

a

,

A

, and

B

, and need to find side

b

. Which law fits?

Sine law
Cosine law
Neither law can use angles.
The law cannot be chosen from the known information.

Show answer and explanation

Sine law

The known values include the opposite pair

a

and

A

. The sine law links that pair to side

b

and its opposite angle

B

.

Question 2

You know two side lengths and the angle between them. Which law is a suitable choice for finding the third side?

Sine law, because it always uses two sides.
Cosine law, because the included angle is known.
Either law without checking the labels.
Neither law can find a side.

Show answer and explanation

Cosine law, because the included angle is known.

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

Question 3

In triangle

ABC

, which side is opposite angle

B

?

Side $a$
Side $b$
Side $c$
There is no matching side.

Show answer and explanation

Side

b

The lowercase side letter matches the capital angle letter that names the opposite angle. Angle

B

is opposite side

b

.

Key terms

Acute angle: An angle measuring less than $90^\circ$ .
Opposite side: The side directly across from a given angle.
Included angle: The angle formed by two specified sides.
Inverse cosine: A calculator function that returns an angle when its cosine value is known.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C1.4. 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.

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