C1.3 Verify the sine law and cosine law using technology

Ontario Grade 11 Mathematics

Trigonometric Functions

Using a dynamic geometry tool to compare measurements and calculations

A triangle gives us two ways to learn about its side lengths and angles. We can measure them in a geometry tool, or calculate relationships using the sine law and cosine law. In this lesson, you will use both approaches on the same triangle and compare the results. The goal is to verify that the laws match the measurements, not to prove them for every possible triangle.

What you will learn

Use a technology tool to construct and measure a triangle.
Check whether side and angle measurements agree with the sine law.
Check whether side and angle measurements agree with the cosine law.
Explain why a check with one triangle is useful but does not prove a rule for every triangle.

1. Prerequisite bridge: sides, angles, and technology

A triangle has three sides and three interior angles. The interior angles are the angles inside the triangle. The three angle measures add to

180^\circ

.

To use the laws, match each lowercase side name to the uppercase angle name across from it. In triangle

ABC

, side

a

is opposite angle

A

, side

b

is opposite angle

B

, and side

c

is opposite angle

C

. This matching matters: using the wrong side-angle pair can make a correct law seem incorrect.

The sine of an angle is a trigonometric ratio. A calculator or geometry tool can give the sine of an angle when the calculator is in degree mode. A dynamic geometry tool lets you make a triangle, move a vertex, and watch measurements change.

Measurements shown by technology are often rounded. For example, a side might display as

7.81

even when its more exact value is not that decimal. Small differences in the last digit are normal when comparing rounded results.

A+B+C=180^\circ

Opposite pairs are $a$ with $A$ , $b$ with $B$ , and $c$ with $C$ .
Use degree mode when entering angle measures in degrees.
Expect small rounding differences.

2. What it means to verify a law

To verify a relationship with technology, first make a triangle and measure its sides and angles. Then substitute those measurements into the relationship. If both sides of an equation are equal, or nearly equal after rounding, the measurements support the relationship for that triangle.

The sine law connects each side with the sine of its opposite angle. Its three ratios should have the same value. This gives a direct check: calculate each ratio and compare the results.

The cosine law connects one side to the other two sides and the cosine of the angle between them. For example, to check side

a

, use sides

b

and

c

with included angle

A

. The included angle is the angle between the two sides used in the calculation.

A single triangle is a useful test, but it cannot establish that a relationship works for every triangle. To build stronger confidence with technology, repeat the check with several different triangles, including triangles with different shapes.

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

The sine law compares three side-to-opposite-angle ratios.
The cosine law uses the angle between the two sides in its calculation.
Verification with technology means checking measured results against a relationship.

3. Two representations of the cosine law

A useful technology check has two representations. The first is a measured triangle on the screen. The second is a numerical calculation using the law. Comparing them helps you see whether the equation describes the shape you built.

The cosine law can be written to check any one of the three sides. For side

a

, the angle used is

A

; for side

b

, use

B

; for side

c

, use

C

. In every case, the angle is opposite the side being checked.

When checking a side, calculate the expression for the square of that side. Then compare it with the square of the measured side. Do not compare a squared result with a side length; the units and values would not match.

A table or the tool's measurement panel can keep the evidence organized. Record each measured side and angle, then record the calculated values. More decimal places can help during calculation, but the final comparison should allow for rounding.

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

Match the target side with its opposite angle.
Compare squared side values when using the squared form of the cosine law.
Record enough digits to make the comparison meaningful.

4. Guided technology check

A coordinate grid makes a clear example. Coordinates tell the tool where to place each vertex. The tool can then draw the triangle and display its side lengths and angle measures. The coordinates are only a way to create the triangle; the verification uses the measured sides and angles.

In the example, the calculated sine ratios are close to one another. The cosine-law calculation also gives the measured side squared. These two comparisons show how technology can check both laws on one triangle.

In your own tool, use its distance and angle measurement features. If your interface names these features differently, use the options that display the length of a segment and the measure of an angle. Keep angle labels visible so you can check that each value is paired with the correct opposite side.

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

Construct first, measure second, and calculate third.
Use the same triangle measurements in both law checks.
A close match is expected; tiny differences can come from rounding.

5. Applying the verification process

For another triangle, follow the same sequence. Draw or construct the triangle in a geometry tool. Measure all three sides and all three angles. Check the sine law by calculating the three ratios. Check the cosine law by selecting a side and using its opposite angle with the other two sides.

You can move a vertex and repeat the checks. The side lengths and angles will change, but the law calculations should continue to agree with the new measurements, subject to display precision. Try a narrow triangle and a wide-looking triangle to see that the check is not limited to one shape.

Watch for three common sources of a mismatch: an angle in the wrong calculator mode, a side paired with an angle that is not opposite it, or too much rounding during intermediate steps. Correct these before deciding that the law failed.

Repeat checks with different triangle shapes.
Keep full calculator precision until the final comparison.
A mismatch is a reason to check setup and labels first.

Keep the side-angle pairs matched

Side	Opposite angle	Sine-law ratio
a	A	a / sin A
b	B	b / sin B
c	C	c / sin C

Worked example

Check both laws on one constructed triangle

In a geometry tool, place

A=(0,0)

,

B=(8,0)

, and

C=(2,5)

. Measure the triangle, then use the measurements to check the sine law and the cosine law for side

a

.

Construct and label
Make triangle $ABC$ from the three points. Label each side with the lowercase letter opposite its matching angle. The tool's measurements, rounded to two decimal places, are approximately $a=7.81$ , $b=5.39$ , and $c=8.00$ . The angles are approximately $A=68.20^\circ$ , $B=39.80^\circ$ , and $C=72.00^\circ$ .
Compare sine-law ratios
Use each measured side with the sine of its opposite angle. The ratios are all about $8.41$ . Their close agreement supports the sine law for this constructed triangle. The slight differences between the displayed ratios come from rounding the measurements.
$\frac{7.81}{\sin 68.20^\circ}\approx\frac{5.39}{\sin 39.80^\circ}\approx\frac{8.00}{\sin 72.00^\circ}\approx 8.41$
Check the cosine law for side a
To check side $a$ , use angle $A$ between sides $b$ and $c$ . The expression gives a value close to $61$ , and the measured side squared is also close to $61$ . This is the expected match, allowing for rounded measurements.
$5.39^2+8.00^2-2(5.39)(8.00)\cos 68.20^\circ\approx 61.00\approx 7.81^2$
Interpret the check
Both calculations agree with the tool's measurements to the displayed precision. This verifies the laws for this example. It does not by itself prove that the laws work for every triangle, so repeat the process with other constructions.

Answer: For this triangle, the three sine-law ratios are approximately

8.41

, and the cosine-law calculation for

a^2

is approximately

61.00

, matching the measured value

7.81^2

.

Check: The coordinates give

AB=8

,

AC=\sqrt{29}\approx5.39

, and

BC=\sqrt{61}\approx7.81

. These values agree with the tool measurements and the comparisons.

Common mistakes and how to avoid them

Pairing side

a

with angle

B

in the sine law.

Correction: Use each side with the angle directly opposite it:

a

with

A

,

b

with

B

, and

c

with C.

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

Correction: For a check of side

a

, use sides

b

and

c

and their included angle

A

.

Treating a tiny rounding difference as evidence that a law is wrong.

Correction: Keep more digits during calculations and compare results at a reasonable level of precision.

Claiming that checking one triangle proves the law for every triangle.

Correction: Describe the result as verification for that example. Repeat with other shapes to gather more evidence.

Lesson summary

A geometry tool can construct triangles and display their side lengths and angle measures.
The sine law is checked by comparing each side-to-sine ratio with the other two.
The cosine law is checked by using the two sides and included angle for the side being tested.
Technology checks support a relationship in the examples tested; one example is not a proof for all triangles.

Check your understanding

Question 1

To check the sine-law ratio for side

b

, which angle should you use?

Angle A
Angle B
Angle C
Any angle, since all are inside the triangle

Show answer and explanation

Angle B

Side

b

is opposite angle

B

, so its ratio is

b/\sin B

.

Question 2

To check the cosine law for side

c

, which information belongs in the calculation?

Sides a and b, with angle C
Sides a and c, with angle B
Sides b and c, with angle A
Sides a and b, with angle A

Show answer and explanation

Sides a and b, with angle C

For side

c

, use the other two sides,

a

and

b

, and the included angle

C

.

Question 3

A technology check gives ratios of 6.24, 6.23, and 6.24 after measurements were rounded. What is the best conclusion?

The sine law is disproved because the ratios are not exactly equal.
The results are close and support the sine law for this triangle, with rounding likely causing the small difference.
Only the largest ratio is valid.
The angle measurements should be ignored.

Show answer and explanation

The results are close and support the sine law for this triangle, with rounding likely causing the small difference.

Rounded measurements can produce small differences. Close ratios support the relationship for the tested triangle.

Key terms

Opposite side: The side directly across from an angle in a triangle.
Included angle: The angle between two named sides.
Verify: Check a relationship against measurements or results. A check supports the relationship in the example tested.
Rounding: Writing a number with fewer digits while keeping it close to its original value.

Continue through MCF3M

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C1.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

C1.3 · Verify the sine law and cosine law using technology

Ontario Grade 11 Mathematics

Using a dynamic geometry tool to compare measurements and calculations

What you will learn

1. Prerequisite bridge: sides, angles, and technology

2. What it means to verify a law

3. Two representations of the cosine law

4. Guided technology check

5. Applying the verification process

Keep the side-angle pairs matched

Check both laws on one constructed triangle

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson