Find acute-triangle sides and angles with sine or cosine law

Ontario Grade 10 Mathematics

Trigonometry

Using the sine law and cosine law

You already know that the three interior angles of a triangle add to

180^\circ

, and that right triangles have special side relationships. But not every triangle is a right triangle. In this lesson, you will use the sine law and cosine law to find missing sides and angles in acute triangles. An acute triangle has three angles that are each less than

90^\circ

. The key is to match each side with the angle directly across from it.

What you will learn

Identify which sides and angles are opposite one another in a triangle.
Choose the sine law or cosine law based on the information given.
Use these laws to find a missing side or angle in an acute triangle.
Check that a calculated angle and side make sense.

1. Review: opposite sides and angles

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

a

is opposite angle

A

. Side

b

is opposite angle

B

, and side

c

is opposite angle

C

. The lowercase letter names a side; the matching uppercase letter names its opposite angle.

This matching matters because both laws use opposite side-angle pairs. Before substituting numbers, label the triangle or write down which side is opposite each angle. The angles in any triangle add to

180^\circ

. This can help you find a third angle if two angles are already known.

A ratio compares two quantities by division. In the sine law, the ratio of a side to the sine of its opposite angle is the same for all three pairs. You may have met sine, written as

\sin

, when studying right triangles. Here, you can use a calculator's sine function with an angle in degree mode.

A+B+C=180^\circ

Match $a$ with $A$ , $b$ with $B$ , and $c$ with $C$ .
An acute angle is greater than $0^\circ$ and less than $90^\circ$ .
Check that your calculator is set to degrees.

2. Choose the law that fits

Use the sine law when you know at least one complete opposite pair: a side and the angle across from it. It is especially useful when you have two angles and one side, or two sides and an opposite angle-side pair. You can set two matching ratios equal and solve for the missing value.

Use the cosine law when you know two sides and the angle between them and need the third side. You can also use it to find an angle when all three sides are known. The angle between two sides is called the included angle. For instance, angle

A

lies between sides

b

and

c

.

A simple choice guide is: look for a known opposite pair for the sine law. If you have two sides and their included angle, use the cosine law to find the third side. If all three sides are known, use the cosine law to find an angle. Keep the triangle labels consistent throughout.

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

Sine law: use corresponding side-angle pairs.
Cosine law: use two sides and their included angle to find a side, or three sides to find an angle.
Write the equation before entering values into a calculator.

3. Use the laws carefully

For the sine law, choose two fractions that contain the known opposite pair and the unknown value. If the unknown is a side, keep the side in the numerator and its opposite-angle sine in the denominator. If the unknown is an angle, isolate its sine and use the inverse sine function on the calculator. The notation

\sin^{-1}

means inverse sine; it finds an angle from a sine value. It is not the same as dividing by sine.

For the cosine law, the side opposite angle

A

is found using sides

b

and

c

and their included angle

A

. To find angle

A

from three sides, rearrange the law to isolate

\cos A

, then use inverse cosine, written

\cos^{-1}

. Keep parentheses around the whole fraction when using a calculator.

Round only after the calculation is complete. If a question does not specify a rounding rule, give a reasonable number of decimal places and include units. Sides are measured in the same units as the given sides; angles are measured in degrees.

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

The side in each sine-law fraction must be paired with its opposite angle.
Use inverse sine or inverse cosine only after isolating the sine or cosine of the unknown angle.
A side length must be positive, and the angles must add to $180^\circ$ .

4. Guided example and independent practice

The example begins with two sides and their included angle. That points to the cosine law for the missing side. Once that side is known, a matching side-angle pair is available, so the sine law can find another angle. The last angle can be checked using the triangle angle sum.

For independent practice, try this: in an acute triangle,

A=42^\circ

,

B=68^\circ

, and

a=7.5

cm. Find

C

and then find

b

. First use the angle sum. Then use the sine law with the known opposite pair

a

and

A

. Check that your resulting angles add to

180^\circ

.

Choose a law from the information you know, not from the value you want to find.
Use the result of one law to create a known opposite pair for the next step when possible.
Check the final angles and units.

Which law should I use?

Known information	Useful law
A side and its opposite angle, plus another side or angle	Sine law
Two sides and the angle between them; find the third side	Cosine law
All three sides; find an angle	Cosine law

Worked example

Find the sides and angles

In an acute triangle,

b=8

cm,

c=9

cm, and

A=50^\circ

. Find side

a

and angles

B

and

C

.

Choose a law for the side
The known sides $b$ and $c$ meet at angle $A$ , so $A$ is the included angle. Use the cosine law to find the opposite side $a$ .
$a^2=b^2+c^2-2bc\cos A$
Substitute and calculate
Substitute the given values. The calculator must be in degree mode. Take the positive square root because a side length is positive.
$a=\sqrt{8^2+9^2-2(8)(9)\cos 50^\circ}\approx 7.24\text{ cm}$
Find angle B
Now side $a$ is known, and $a$ is opposite the known angle $A$ . Use that pair with side $b$ to find angle $B$ . Since the triangle is acute, $B$ must be less than $90^\circ$ .
$\sin B=\frac{8\sin 50^\circ}{7.24},\qquad B\approx 57.8^\circ$
Find and check angle C
Use the triangle angle sum to find the remaining angle. Check that the result is acute and that all three angles total $180^\circ$ .
$C=180^\circ-50^\circ-57.8^\circ=72.2^\circ$

Answer: The missing side is approximately

7.24

cm. The missing angles are approximately

B=57.8^\circ

and

C=72.2^\circ

.

Check: All three angles are less than

90^\circ

, as required for an acute triangle. Their sum is

50^\circ+57.8^\circ+72.2^\circ=180^\circ

. The side lengths also fit their opposite angles: the largest angle,

C

, is opposite the largest side,

c=9

cm.

Common mistakes and how to avoid them

Pairing a side with an angle beside it instead of the angle opposite it.

Correction: Check the triangle labels:

a

is opposite

A

,

b

is opposite

B

, and

c

is opposite

C

.

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

Correction: For the side form of the cosine law, use the angle between the two sides in the equation.

Treating

\sin^{-1}

as

1/\sin

.

Correction: Use the calculator's inverse sine function to find an angle from a sine value.

Rounding early or leaving the calculator in radian mode.

Correction: Set the calculator to degrees and keep extra digits until the final answer.

Lesson summary

Match every side with the angle directly opposite it.
Use the sine law when you have a known opposite side-angle pair.
Use the cosine law with two sides and their included angle to find a side, or with three sides to find an angle.
Use the angle sum and the acute-triangle condition to check your results.

Check your understanding

Question 1

In triangle

ABC

, which angle is opposite side

c

?

$A$
$B$
$C$
correctIndex: 2,

Show answer and explanation

C

Side

c

is opposite angle

C

; matching uppercase and lowercase labels identify opposite pairs.

Question 2

You know two sides and the angle between them. Which law is the best choice to find the third side?

Sine law
Cosine law
The triangle angle sum alone
correctIndex: 1,

Show answer and explanation

Cosine law

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

Question 3

An acute triangle has

A=40^\circ

and

B=75^\circ

. What is

C

?

$55^\circ$
$65^\circ$
$105^\circ$
correctIndex: 1,

Show answer and explanation

65^\circ

The angles add to

180^\circ

, so

C=180^\circ-40^\circ-75^\circ=65^\circ

.

Key terms

Opposite: Across from a side or angle in a triangle.
Included angle: The angle between two specified sides.
Sine law: A rule stating that each side divided by the sine of its opposite angle gives the same ratio.
Cosine law: A rule connecting a triangle's sides with the cosine of one of its angles.
Inverse sine: A calculator function that finds an angle when its sine value is known.
Inverse cosine: A calculator function that finds an angle when its cosine value is known.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T9. 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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