C2.2 Verify the sine law and cosine law using technology

Ontario Grade 11 Mathematics

Geometry and Trigonometry

MBF3C study topic C2.2: compare triangle measurements with the laws’ predictions

A ruler and protractor can give approximate measurements of a triangle. A digital tool can measure the same triangle and calculate values quickly. The sine law and cosine law describe relationships that should hold among a triangle’s sides and angles. To verify a law, use technology to compare what the law predicts with measurements from a triangle. A successful check supports the relationship for that triangle. It does not, by itself, prove the law for every triangle.

What you will learn

Match each triangle side with its opposite angle.
Use a calculator or digital tool to check the sine law by comparing three ratios.
Use a calculator or digital tool to check the cosine law by comparing a predicted side value with a measured value.
Explain why displayed values may differ slightly because of rounding.

1. Review: label and measure a triangle

A triangle has three sides and three interior angles. An interior angle is an angle inside the triangle. We use uppercase letters such as

A

,

B

, and

C

for the angles. The matching lowercase letters

a

,

b

, and

c

name the sides opposite those angles. Opposite means directly across from an angle, not touching it.

For example, side

a

joins the corners

B

and

C

, so it is across from angle

A

. This matching is important when checking either law. A side paired with the wrong angle can make a correct law appear not to work.

A ruler, protractor, calculator, spreadsheet, or dynamic geometry tool can help with a check. A dynamic geometry tool lets you change a triangle and see updated measurements. Record side lengths in one consistent unit, such as centimetres. When entering angles shown in degrees, set the calculator to degree mode. Measurements and displayed decimals are approximate, even when a tool shows many digits.

A\leftrightarrow a,\quad B\leftrightarrow b,\quad C\leftrightarrow c

Match each uppercase angle with its opposite lowercase side.
Use consistent units for side lengths and degree mode for angles.
Treat measured and displayed values as approximate.

2. Verify the sine law with ratios

The sine of an angle is the number a calculator gives when you use its sine function. For this check, enter the angle in degree mode. The sine law says that dividing each side by the sine of its opposite angle gives the same ratio.

To check the law, first label or record all three opposite side-and-angle pairs. Use a calculator or spreadsheet to find the sine of each angle. Then divide the matching side by that sine value. Compare the three results. If the law fits the measured triangle, the ratios should be equal or close to equal.

For instance, a spreadsheet could hold each side and angle in a separate cell and calculate the three quotients. A quotient is the result of a division. A dynamic geometry tool can also show the measurements while you change a vertex. The side lengths and angles will change, but the ratios should continue to be close.

The final digits may differ because the measurements or calculator display have been rounded. Compare to a sensible number of decimal places rather than expecting every displayed digit to match. A close result verifies the relationship for the triangle you checked; it is not a proof for all triangles.

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

Divide each side by the sine of its opposite angle.
Compare all three ratios.
Allow for small differences caused by measurement and rounding.

3. Verify the cosine law by comparing a side

The cosine law connects the lengths of three sides with one angle. For a direct check, choose two known sides and the angle where they meet. This angle is called the included angle. The third side is opposite the included angle.

Use the cosine law to calculate a predicted value for the square of the opposite side. A calculator can evaluate the expression. Then compare that prediction with the square of the side measured by the tool. The values should be equal or close, allowing for rounding.

For example, a spreadsheet can calculate the predicted squared side from two measured sides and their included angle. A second cell can square the measured opposite side. Comparing those two cells checks the law without needing to rearrange it to find an unknown side.

If the values are not close, check that you used the angle between the two known sides and the side opposite that angle. Also check the labels and calculator mode. These simple checks can catch common input errors.

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

Use the angle where the two known sides meet.
Compare the predicted squared side with the measured side squared.
Check labels and calculator mode if the values differ.

4. Make the technology check useful

Start with a clearly labelled triangle. Record its measurements and the precision shown by the tool. For the sine law, calculate all three side-over-sine ratios. For the cosine law, calculate the predicted squared side and compare it with the square of the measured side. Keep enough digits during the calculations, then compare the results at a reasonable precision.

Repeating the check with another triangle gives more evidence that the relationship is consistent. In a dynamic geometry tool, change the triangle’s shape and watch the measurements update. Avoid flattening the triangle, since the check is meant to use a triangle with a clear interior shape.

Technology can measure and calculate, but the user must choose the correct labels and inputs. If a result looks wrong, check the opposite-side pairing, the included angle, units, and degree mode. Small differences may simply come from rounded measurements.

Use a clear triangle and record the tool’s displayed precision.
Repeat the check with another triangle when possible.
Interpret small differences in light of rounding and measurement.

Worked example

Example 1: Check the sine law

A geometry tool displays a triangle with angles

A=30^\circ

,

B=60^\circ

, and

C=90^\circ

. The opposite sides are

a=5

,

b\approx8.660

, and

c=10

centimetres. Use a calculator to check the sine law.

Match the pairs
The labels pair each angle with the side directly across from it. Keep the lengths in centimetres and use degree mode for the sine calculations.
Calculate the ratios
Divide each side by the sine of its opposite angle. The results are close to 10; the middle result uses a rounded side measurement.
$\frac{5}{\sin 30^\circ}=10,\quad \frac{8.660}{\sin 60^\circ}\approx10.000,\quad \frac{10}{\sin 90^\circ}=10$
Compare the results
All three ratios agree to the displayed precision. This verifies the sine-law relationship for this measured triangle.

Answer: The three ratios are approximately 10 centimetres, so the measured triangle agrees with the sine law.

Check: Since

\sin 30^\circ=0.5

, the first ratio is

5\div0.5=10

. Since

\sin 90^\circ=1

, the third ratio is 10. The rounded middle measurement gives approximately 10 as well.

Worked example

Example 2: Check the cosine law

A geometry app shows two sides as

a=5

cm and

b=7

cm, with included angle

C=60^\circ

. The app measures the opposite side as approximately

c=6.245

cm. Use a calculator to check the cosine law.

Identify the values to compare
Side $c$ is opposite angle $C$ . The law predicts $c^2$ from the two known sides and their included angle, so compare that prediction with the measured side squared.
Calculate the predicted value
Substitute the measurements and use degree mode. The cosine of $60^\circ$ is $0.5$ , giving a predicted squared side of 39 square centimetres.
$c^2=5^2+7^2-2(5)(7)\cos 60^\circ=39$
Compare with the app measurement
Square the app’s displayed length. This result is approximately 39, which agrees with the predicted value to the displayed precision.
$(6.245)^2\approx39.000$

Answer: The predicted squared side is

39

square centimetres, and the measured side squared is approximately

39

. The triangle agrees with the cosine law.

Check: The predicted side length is approximately

6.245

cm because its square is 39. This matches the app’s displayed measurement.

Common mistakes and how to avoid them

Pairing a side with an angle that touches it when checking the sine law.

Correction: Use the angle directly opposite the side. Check the uppercase and lowercase labels before calculating.

Using an angle other than the included angle in the cosine-law check.

Correction: Use the angle where the two known sides meet. The side being checked is opposite that angle.

Treating a small difference in displayed decimals as evidence that a law fails.

Correction: Check measurement precision and rounding. Compare values to a reasonable number of decimal places.

Entering degree measures while the calculator is not set to degree mode.

Correction: Set the calculator to degree mode before calculating a sine or cosine from angles shown in degrees.

Lesson summary

For the sine law, compare the three ratios formed by dividing each side by the sine of its opposite angle.
For the cosine law, compare the predicted squared side with the square of the measured side.
Correct labels, units, and degree mode are needed for a useful technology check.
Small differences may result from measurement and rounding.

Check your understanding

Question 1

A triangle has

A=45^\circ

,

B=45^\circ

,

C=90^\circ

, and opposite sides

a=4

,

b=4

,

c\approx5.657

. Which set of calculations checks the sine law?

$\frac{4}{\sin45^\circ}$ , $\frac{4}{\sin45^\circ}$ , and $\frac{5.657}{\sin90^\circ}$
$\frac{4}{\sin90^\circ}$ , $\frac{4}{\sin45^\circ}$ , and $\frac{5.657}{\sin45^\circ}$
$4\sin45^\circ$ , $4\sin45^\circ$ , and $5.657\sin90^\circ$
$\frac{4}{\sin45^\circ}$ , $\frac{4}{\sin90^\circ}$ , and $\frac{5.657}{\sin45^\circ}$

Show answer and explanation

\frac{4}{\sin45^\circ}

,

\frac{4}{\sin45^\circ}

, and

\frac{5.657}{\sin90^\circ}

Each side must be divided by the sine of its opposite angle. The three ratios are approximately

5.657

.

Question 2

For sides

a=3

cm and

b=4

cm with included angle

C=90^\circ

, what does the cosine-law check predict for

c^2

?

$25$ square centimetres
$7$ square centimetres
$49$ square centimetres
$24$ square centimetres

Show answer and explanation

25

square centimetres

Substitution gives

3^2+4^2-2(3)(4)\cos90^\circ=25

, since

\cos90^\circ=0

.

Question 3

A technology check gives sine-law ratios of 8.00, 8.01, and 7.99. What is the best conclusion?

The values are close, so rounding or measurement precision likely explains the small differences.
The sine law is false because the ratios are not exactly identical.
The values must be added before deciding whether the relationship holds.
The triangle’s side labels should be changed until all values are identical.

Show answer and explanation

The values are close, so rounding or measurement precision likely explains the small differences.

Measured and displayed values are often rounded. Ratios this close provide a reasonable verification for the measured triangle.

Key terms

Opposite side: The side directly across from a chosen angle in a triangle.
Included angle: The angle formed where two specified sides meet.
Quotient: The result of dividing one number by another.
Verify: Check a relationship by comparing its prediction with measured or calculated values.
Rounding: Writing a number with fewer digits, which can make it slightly different from the unrounded value.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation C2.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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C2.2 · Verify the sine law and cosine law using technology

Ontario Grade 11 Mathematics

MBF3C study topic C2.2: compare triangle measurements with the laws’ predictions

What you will learn

1. Review: label and measure a triangle

2. Verify the sine law with ratios

3. Verify the cosine law by comparing a side

4. Make the technology check useful

Example 1: Check the sine law

Example 2: Check the cosine law

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MBF3C

About this lesson