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

Ontario Grade 11 Mathematics

Geometry and Trigonometry

Grade 11 MBF3C — C2.3: Choose and apply a law in acute triangles

The sine law and cosine law help find missing measurements in triangles that are not necessarily right triangles. This lesson focuses on acute triangles. An acute triangle has three angles, each less than

90^\circ

. Your first decision is to look at the information given and choose the law that uses it. Before calculating, review how sides and angles are labelled: in a triangle labelled

A

,

B

, and

C

, the lowercase side

a

is opposite angle

A

, and so on.

What you will learn

Identify opposite side-angle pairs in an acute triangle.
Choose the sine law or cosine law based on the measurements provided.
Apply the selected law to find a missing side or angle.
Check that a result is reasonable for an acute triangle.

1. Review: label the triangle and its measurements

An angle is the opening formed by two sides. The side opposite an angle is directly across from it. For example, side

a

is opposite angle

A

. These matching pairs are important when using the sine law.

The angles inside any triangle add to

180^\circ

. If two angles are known, subtract their sum from

180^\circ

to find the third. An acute triangle has all three angles below

90^\circ

. A diagram may not be drawn to scale, so use the measurements provided rather than estimating from its appearance.

A+B+C=180^\circ

Match each uppercase angle with the lowercase side directly opposite it.
The interior angles of a triangle total $180^\circ$ .

2. Choose the law that fits the known information

The sine law compares opposite side-angle pairs. Choose it when you have a known pair, such as angle

A

and opposite side

a

, and enough other information to find the requested measurement. For example, two known angles and one side let you find another side. If necessary, first find the third angle using the triangle angle total.

The cosine law connects the three sides of a triangle with one of its angles. Use it to find a third side when you know two sides and the angle between them. That angle is called the included angle: it is where the two known sides meet. You can also use the cosine law to find an angle when all three sides are known.

A useful routine is to list the known sides and angles, identify any opposite side-angle pair, and check which law uses the information you have. Do not choose a law just because it looks familiar. Match the law to the measurements.

After calculating, check the result. Side lengths must be positive. Each angle in an acute triangle must be less than

90^\circ

, and all three angles must total

180^\circ

. Keep extra calculator digits during the calculation and round at the end.

\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 to find a third side, or three sides to find an angle.
Check the answer against the requirements for an acute triangle.

3. Apply a law and check the result

When using the sine law, keep each side matched with its opposite angle. If you are finding an angle, use the inverse sine function on a calculator. Set the calculator to degrees because the angles in this lesson are measured in degrees.

When using the cosine law to find a missing side, the angle in the calculation must be between the two known sides. When finding an angle from three known sides, make sure the side opposite the target angle is in the matching position in the formula.

Keep side units consistent. If the given lengths are in metres, the calculated length is also in metres. A practical model might use two measured paths from one meeting point and the angle between them to find the direct distance between their endpoints. The calculated distance should fit the known lengths and angle.

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

Use degree mode for angle calculations in this lesson.
Include units with side-length answers.
A reasonable answer fits the known triangle measurements.

Match the measurements to a law

Known information	Useful law	Why it fits
An opposite side-angle pair and enough other information	Sine law	It compares opposite side-angle pairs.
Two sides and their included angle; find a third side	Cosine law	It uses the two sides and the angle between them.
All three sides; find an angle	Cosine law	It relates the three sides to the angle opposite the target angle.

Worked example

Example 1: Find a side using the sine law

In an acute triangle,

A=42^\circ

,

B=68^\circ

, and side

a=9.5

cm. Find side

b

to the nearest tenth of a centimetre.

Identify the opposite pair
Side $a$ is opposite angle $A$ , so the known opposite pair is $9.5$ cm and $42^\circ$ . The sine law relates this pair to side $b$ and its opposite angle, $B$ .
$\frac{a}{\sin A}=\frac{b}{\sin B}$
Substitute the measurements
Replace the labels with their given values. Keeping the opposite pairs together gives an equation with only $b$ unknown.
$\frac{9.5}{\sin 42^\circ}=\frac{b}{\sin 68^\circ}$
Solve for the side
Multiply both sides by $\sin 68^\circ$ to leave $b$ by itself. Evaluate the expression in degree mode and round the final result.
$b=\frac{9.5\sin 68^\circ}{\sin 42^\circ}\approx 13.1\text{ cm}$

Answer: Side

b

is approximately

13.1

cm.

Check: Angle

B

is larger than angle

A

, so its opposite side should be longer than side

a

. The third angle is

70^\circ

, making all three angles acute.

Worked example

Example 2: Find a side using the cosine law

Two sides of an acute triangle are

7.2

m and

10.0

m. Their included angle is

54^\circ

. Find the side opposite the

54^\circ

angle to the nearest tenth of a metre.

Choose the law
Two sides and the angle between them are known, so the cosine law directly gives the third side. Let $c$ be opposite the included angle, $C$ .
$c^2=a^2+b^2-2ab\cos C$
Substitute the measurements
Use the two known lengths for $a$ and $b$ , and the included angle for $C$ .
$c^2=7.2^2+10.0^2-2(7.2)(10.0)\cos 54^\circ$
Calculate the side length
Evaluate the expression in degree mode, then take the positive square root because a side length is positive.
$c=\sqrt{7.2^2+10.0^2-2(7.2)(10.0)\cos 54^\circ}\approx 8.3\text{ m}$

Answer: The side opposite the

54^\circ

angle is approximately

8.3

m.

Check: The result is positive and shorter than either known side. This is reasonable for the given acute included angle.

Common mistakes and how to avoid them

Pairing an angle with a side beside it in the sine law.

Correction: Use the side directly opposite the angle. Side

a

pairs with angle

A

.

Using an angle that is not between the two known sides when finding a third side with the cosine law.

Correction: Use the included angle: the angle where the two known sides meet.

Using calculator settings that do not match angles measured in degrees.

Correction: Set the calculator to degrees before evaluating sine or cosine.

Rounding intermediate values too early.

Correction: Keep the calculator value through the calculation and round the final answer to the requested precision.

Lesson summary

Pair each lowercase side with its opposite uppercase angle.
Choose the sine law when the information includes an opposite side-angle pair.
Choose the cosine law for two sides and their included angle, or for three known sides when finding an angle.
Check units and whether the result fits an acute triangle.

Check your understanding

Question 1

An acute triangle has

A=35^\circ

,

B=75^\circ

, and

a=8

cm. Which law is the most direct choice for finding

b

?

Sine law
Cosine law, because two sides are known
Cosine law, because all three sides are known
Triangle angle total alone

Show answer and explanation

Sine law

The known side

a

is opposite the known angle

A

. That opposite pair makes the sine law the direct choice.

Question 2

Two sides are

6

cm and

9

cm, and their included angle is

48^\circ

. Which law directly finds the third side?

Sine law
Cosine law
Triangle angle total alone
Opposite side-angle pairing alone

Show answer and explanation

Cosine law

Two sides and their included angle are the information needed to use the cosine law to find the third side.

Question 3

A triangle calculation gives angles of

41^\circ

,

63^\circ

, and

76^\circ

. Is the result consistent with an acute triangle?

Yes; all three are less than $90^\circ$ and total $180^\circ$ .
No; the angles total more than $180^\circ$ .
No; an acute triangle must have one angle equal to $90^\circ$ .
No; one angle is greater than $90^\circ$ .

Show answer and explanation

Yes; all three are less than

90^\circ

and total

180^\circ

.

The angles total

180^\circ

, and each is less than

90^\circ

, so they are consistent with an acute triangle.

Key terms

Opposite side: The side directly across from a specified angle.
Included angle: The angle where two specified sides meet.
Acute triangle: A triangle with all three angles less than $90^\circ$ .
Sine law: A rule that relates each side of a triangle to the sine of its opposite angle.
Cosine law: A rule that relates a side to the other sides and the angle between them.

Continue through MBF3C

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation C2.3. 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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