Cosine Law for Non-Right Triangles

Alberta Grade 11 Mathematics

Geometry

Choosing and using the sine law and cosine law

Triangles are used in surveying, navigation, and design. Sometimes a few measured sides and angles are enough to work out the rest. The sine law and cosine law connect these measurements. This lesson focuses on problems where the given information determines one triangle. It does not use the ambiguous case, where measurements can allow more than one triangle.

What you will learn

Match each angle with its opposite side.
Choose the sine law or cosine law to suit the information given.
Find unknown sides and angles in a triangle with one solution.
Check that answers fit the triangle and use appropriate units.

1. Review: angles, sides, and opposite pairs

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

180^\circ

. A side is opposite an angle when it lies directly across from that angle.

We label the angles

A

,

B

, and

C

. We label the opposite sides with lowercase letters: side

a

is opposite angle

A

, side

b

is opposite angle

B

, and side

c

is opposite angle

C

. These pairs matter because the sine law uses each angle with its opposite side.

An included angle is the angle between two sides. For example, if two known sides meet at angle

C

, then

C

is their included angle. Two sides and their included angle are called SAS information. Two angles and a side are called ASA or AAS information. These patterns determine one triangle.

A+B+C=180^\circ

Match $A$ with $a$ , $B$ with $b$ , and $C$ with $c$ .
The interior angles of a triangle add to $180^\circ$ .
Use degree mode when the angles are given in degrees.

2. Choose the law that fits

The sine law links an angle to its opposite side. Use it when you have at least one complete opposite pair: a known angle and the side across from it. With ASA or AAS information, first find any missing angle using the angle sum. Then use the sine law to find unknown sides.

The cosine law links three sides to an angle. Use it for SAS information to find the side opposite the included angle. You can also use it when all three sides are known and you need an angle.

A useful first question is: what information is given? Two sides and their included angle point to the cosine law. Two angles and a side point to the sine law after you find the third angle. Three sides point to the cosine law if an angle is unknown.

Keep side lengths in the same unit. Carry extra calculator digits through your work and round the final answer to a sensible precision. The calculator’s inverse-sine key, often shown as

\sin^{-1}

, finds an angle when you know its sine value. Use it only after forming the correct sine ratio.

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

Use the sine law when a known opposite angle-side pair is available.
Use the cosine law for SAS or to find an angle from three sides.
Check that the calculator is set to degrees.

3. The cosine law for SAS

Suppose the known sides are

a

and

b

, and the angle between them is

C

. The side across from that angle is

c

. The cosine law uses the two known sides and their included angle to find

c

.

For example, take sides of

7

m and

10

m with an included angle of

60^\circ

. Substituting gives

c^2=79

, so the positive length is about

8.89

m. A side length must be positive. The diagram in the first worked example shows these measurements and the unknown side.

The angle must be between the two sides used in the formula. Once you have the third side, you have a known opposite pair: the new side and its included angle. You can then use the sine law to find another angle.

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

The cosine-law angle is the included angle between the two known sides.
Use the positive square root for a side length.
The calculated side has the same unit as the other sides.

4. Use the sine law and check the result

After finding a side with the cosine law, use the sine law to find an angle. For example, if

a

is opposite

A

and

c

is opposite

C

, the matching ratios use

a

with

A

and

c

with

C

. Rearrange the ratio to find the sine of the unknown angle, then use the calculator’s inverse-sine function.

Find the last angle by subtracting the known angles from

180^\circ

. This gives a direct check: all three angles should be positive and add to

180^\circ

.

Compare side sizes with their opposite angles. The longest side should be opposite the largest angle, and the shortest side should be opposite the smallest angle. This is a useful reasonableness check, not a replacement for the laws.

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

Keep opposite side-angle pairs together in every sine-law ratio.
Use the triangle angle sum to find and check the last angle.
Check side units, positive lengths, and the relationship between side and angle sizes.

Worked example

Find a third side

Two sides of a triangular garden are 7 m and 10 m. The angle between them is

60^\circ

. Find the third side.

Identify the information
The two known sides meet at the given angle. This is SAS information, so use the cosine law to find the side opposite the included angle.
$a=7,\quad b=10,\quad C=60^\circ$
Substitute into the cosine law
The unknown side is opposite angle $C$ . Substitute the known side lengths and angle.
$c^2=7^2+10^2-2(7)(10)\cos 60^\circ=79$
Find the positive length
A side length is positive, so take the positive square root. Round to the nearest hundredth of a metre.
$c=\sqrt{79}\approx8.89\text{ m}$

Answer: The third side is approximately 8.89 m.

Check: The result is greater than the difference of the known sides and less than their sum. It is a reasonable third side.

Worked example

Find the side and remaining angles

A triangular sign has two sides of 9 cm and 12 cm with an included angle of

40^\circ

. Find the third side and the other two angles.

Find the third side
The given information is SAS. Let $C$ be the included angle and $c$ the side opposite it. Use the cosine law.
$c^2=9^2+12^2-2(9)(12)\cos40^\circ\approx59.53$
Take the positive square root
A side length cannot be negative. Keep extra digits for the next calculation.
$c\approx7.72\text{ cm}$
Find the angle opposite the 9 cm side
The known opposite pair is $c$ and $C$ . Use it with the 9 cm side to find angle $A$ .
$\sin A=\frac{9\sin40^\circ}{7.7159},\quad A\approx48.6^\circ$
Find the last angle
The angles of a triangle add to $180^\circ$ . Subtract the two known angles.
$B=180^\circ-40^\circ-48.6^\circ\approx91.4^\circ$

Answer: The third side is about 7.72 cm. The other angles are about

48.6^\circ

and

91.4^\circ

.

Check: The angles add to

180^\circ

. The smallest side, about 7.72 cm, is opposite the smallest angle,

40^\circ

.

Worked example

Solve with an obtuse included angle

A triangular plot has sides of 5.5 km and 8 km with an included angle of

110^\circ

. Find the third side and the two remaining angles.

Use the cosine law
The two known sides include the $110^\circ$ angle. Use the cosine law to find the side opposite that angle.
$c^2=5.5^2+8^2-2(5.5)(8)\cos110^\circ\approx124.35$
Find the side length
Take the positive square root. Since the included angle is obtuse, the opposite side should be the longest side.
$c\approx11.15\text{ km}$
Find the angle opposite 5.5 km
Use the sine law with the known opposite pair $c$ and $C$ to find angle $A$ .
$\sin A=\frac{5.5\sin110^\circ}{11.151},\quad A\approx27.6^\circ$
Find the remaining angle
Subtract the known angles from $180^\circ$ to find the last angle.
$B=180^\circ-110^\circ-27.6^\circ\approx42.4^\circ$

Answer: The third side is about 11.15 km. The remaining angles are about

27.6^\circ

and

42.4^\circ

.

Check: The angles add to

180^\circ

. The longest side is opposite the largest angle,

110^\circ

.

Common mistakes and how to avoid them

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

Correction: For SAS, identify the included angle. It must be between the two sides substituted into the formula.

Pairing an angle with a side that is not opposite it.

Correction: Write down the pairs

A

with

a

,

B

with

b

, and

C

with

c

before using the sine law.

Using the sine law before a known opposite pair is available.

Correction: With SAS information, first use the cosine law to find the third side. Then you have a known opposite pair.

Rounding intermediate values too early.

Correction: Keep extra calculator digits during your work. Round the final answers to a sensible precision.

Lesson summary

Match every angle with its opposite side.
Use the cosine law for two sides and their included angle, or to find an angle when three sides are known.
Use the sine law when a known opposite side-angle pair is available.
Check positive side lengths, units, and the $180^\circ$ angle sum.

Check your understanding

Question 1

Two sides are 6 cm and 8 cm, and their included angle is

90^\circ

. Which law should you use first to find the third side?

The cosine law
The sine law, because the included angle is known
The triangle angle sum
No law is needed

Show answer and explanation

The cosine law

Two sides and their included angle are SAS information, so begin with the cosine law.

Question 2

In a triangle,

A=50^\circ

,

B=70^\circ

, and side

a=9

m. What is angle

C

?

$40^\circ$
$60^\circ$
$70^\circ$
$110^\circ$

Show answer and explanation

60^\circ

The angles add to

180^\circ

, so

C=180^\circ-50^\circ-70^\circ=60^\circ

.

Key terms

Opposite side: The side directly across from a named angle.
Included angle: The angle between two given sides.
SAS: Two sides and the angle between them.
Sine law: A rule relating each triangle angle to its opposite side.
Cosine law: A rule relating a triangle’s sides to one of its angles.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation G3. 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

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G3 · Cosine Law for Non-Right Triangles

Alberta Grade 11 Mathematics

Choosing and using the sine law and cosine law

What you will learn

1. Review: angles, sides, and opposite pairs

2. Choose the law that fits

3. The cosine law for SAS

4. Use the sine law and check the result

Find a third side

Find the side and remaining angles

Solve with an obtuse included angle

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Math 20-2

Tutor support for G3 · Solve non-ambiguous triangle problems with sine and cosine laws

About this lesson and its review