Cosine and Sine Rules
Cosine and Sine Rules
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Cosine and Sine Rules
Open with the idea that right-angled triangle methods do not solve every triangle. This lesson extends triangle solving to any triangle when the given information fits the sine rule or cosine rule.
Learning goal
I can use the sine rule and cosine rule to solve triangle problems.
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What words come to mind when you hear sine rule or cosine rule?
Use this to surface prior knowledge and misconceptions such as 'right-angled', 'opposite', 'angle', 'side', 'formula' or 'SOHCAHTOA'.
When do we need these rules?
Pythagoras and basic trigonometric ratios work only in right-angled triangles.
For any triangle, we use:
the sine rule when we have a known side-angle pair
the cosine rule when we need a missing side from two sides and an included angle, or a missing angle from three sides
Stress the decision-making idea early: students often know the formulas but choose the wrong one. A side-angle pair means a side and its opposite angle.
Naming sides and angles
In triangle notation:
side a is opposite angle A
side b is opposite angle B
side c is opposite angle C
Matching each side to its opposite angle correctly is essential before using either rule.
Remind students that formula errors often begin with poor labelling, not algebra. Encourage them to sketch and label first, even if a diagram is given.
The sine rule
The sine rule links each side to the sine of its opposite angle:
a / sin A = b / sin B = c / sin C
Use it when:
you know a side and its opposite angle
you want another side or angle
the triangle is not right-angled
You may also mention the rearranged forms a/sinA = b/sinB and sinA/a = sinB/b are equivalent, but keep one consistent form on the board.
The cosine rule
The cosine rule is:
a² = b² + c² - 2bc cos A
Similar forms exist for b² and c².
Use it when:
two sides and the included angle are known
all three sides are known and an angle is needed
Connect it briefly to Pythagoras: if A = 90°, cos A = 0, so the rule becomes a² = b² + c².
Choosing the correct rule
A quick guide:
Side, opposite angle, and another side/angle → usually sine rule
Two sides and included angle → cosine rule
Three sides → cosine rule first
Always check what information is given before substituting values.
Students often try the sine rule just because they see an angle. Emphasise that the presence of a matching opposite pair is the key sign.
Which clue usually tells you to use the sine rule?
Three sides known
An opposite pair
Included angle
Right angle only
Tests the decision rule from slides 2 to 6. Students often choose an option just because an angle is present; the key idea is a known side and its opposite angle.
Complete the summary
For any triangle, choose the rule from the information given. If you know a side and its opposite angle, the sine rule is usually the best start. If you know two sides and the included angle, use the cosine rule.
All blanks are load-bearing terms taught in the first subtopic.
In triangle notation, which side is opposite angle B?
a
b
c
A
Checks correct matching of labels before students apply either formula.
Example 1: find a side
Given A = 40°, B = 65°, a = 8 cm. Find b.
Using the sine rule:
8 / sin 40° = b / sin 65°
So:
b = 8 × sin 65° / sin 40° ≈ 11.3 cm
Point out that this is the most direct sine-rule case because a complete opposite pair is given: a with A.
Example 2: find an angle
Given a = 12 m, b = 9 m, B = 35°. Find A.
Using the sine rule:
12 / sin A = 9 / sin 35°
So:
sin A = 12 × sin 35° / 9 ≈ 0.765
A ≈ 49.9°
This is a good point to mention that inverse sine can sometimes lead to an ambiguous second angle, though this example gives a valid acute result and the lesson does not need to dwell on the full ambiguous case.
Example 3: find the third angle first
Given A = 52°, C = 71°, a = 10 cm. Find c.
First find B only if needed for context:
B = 180° - 52° - 71° = 57°
Then use the sine rule:
10 / sin 52° = c / sin 71°
c ≈ 12.0 cm
Model good habits: if two angles are known, students can still go straight to the needed opposite pair without using the third angle, but checking the triangle sum reinforces structure.
Example 4: another sine rule side
Given B = 48°, C = 77°, b = 15 cm. Find c.
Using the sine rule:
15 / sin 48° = c / sin 77°
So:
c = 15 × sin 77° / sin 48° ≈ 19.7 cm
Keep the worked structure consistent: identify the opposite pair, write the rule, substitute, rearrange, round.
For a = 12, b = 9 and B = 35°, what do you find first?
Side c
Angle A
Angle C
Area
This echoes Example 2. Because b and B form a known opposite pair, the sine rule can be used directly to find A.
Complete the worked-method summary
In the sine-rule examples, first match each side to its opposite angle. Then write a pair such as 8 / sin 40° = b / sin 65°. To find a missing side, rearrange and calculate. To find a missing angle, use inverse sine after isolating the sine value.
Reinforces the repeated method structure from Examples 1 to 4.
What is the first thing you check before using the sine rule?
Expected points: identify the known opposite pair first, choose the sine rule, substitute carefully, and for angle-finding use inverse sine. Students may also mention checking labels and units.
Using the cosine rule for a side
If two sides and the included angle are known, use:
a² = b² + c² - 2bc cos A
This is useful because there is no known opposite side-angle pair, so the sine rule cannot start the problem.
This slide marks the shift from sine-rule situations to cosine-rule situations. Make the contrast explicit.
Example 5: find a side
Given b = 7 cm, c = 11 cm, A = 58°. Find a.
a² = 7² + 11² - 2(7)(11)cos 58°
a² ≈ 88.4
a ≈ 9.4 cm
Students often forget to square-root at the end. Also remind them that the angle used must be the included angle between the two known sides.
Example 6: larger values
Given b = 13 m, c = 17 m, A = 102°. Find a.
a² = 13² + 17² - 2(13)(17)cos 102°
a² ≈ 549.0
a ≈ 23.4 m
A useful check: because A is obtuse, the side opposite it should be the longest side. The result 23.4 m fits that expectation.
When two sides and the included angle are known, which rule fits?
Sine rule
Cosine rule
Pythagoras
Triangle sum
Checks the shift at slide 11 into cosine-rule side problems.
Complete the cosine-rule summary
When there is no known opposite side-angle pair, the sine rule cannot start the problem. If two sides and the included angle are known, use the cosine rule to find the missing side. After calculating a², remember to take the square root to get the final length.
Targets two common issues from slides 11 to 13: choosing the wrong rule and forgetting the final square root.
Which check helps most in Example 6 with A = 102°?
Opposite pair
Longest side
Angles add to 180°
Unit conversion
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Using the cosine rule for an angle
If all three sides are known, rearrange the cosine rule:
cos A = (b² + c² - a²) / 2bc
Then use inverse cosine to find A.
This method finds an angle when no angle is given at the start.
Warn students not to use inverse sine here. With three sides known, cosine rule is the correct starting method.
Example 7: find an angle
Given a = 14 cm, b = 10 cm, c = 9 cm. Find A.
cos A = (10² + 9² - 14²) / 2(10)(9)
cos A = -15 / 180 = -0.0833
A ≈ 94.8°
Again, the answer is obtuse because cosine is negative. This is a good numerical sense check.
Example 8: find another angle
Given a = 6 m, b = 8 m, c = 11 m. Find C.
cos C = (6² + 8² - 11²) / 2(6)(8)
cos C = -21 / 96 = -0.21875
C ≈ 102.6°
Use this final example to reinforce that the letter in the formula must match the side opposite the angle being found.
Common checks for accuracy
After solving a triangle, check that:
each side matches its opposite angle logically
the largest angle is opposite the longest side
angles add to 180°
answers are rounded sensibly for the question
This consolidates method and sense-checking. It helps students catch impossible answers without redoing the whole problem.
If all three sides are known and you need an angle, what do you use first?
Inverse sine
Cosine rule
Sine rule
Pythagoras
Directly targets the warning on slide 14 that students should not start with inverse sine when all three sides are given.
Complete the angle-finding method
When all three sides are known, rearrange the cosine rule to find cos A. Then use inverse cosine to get the angle. A useful check is that the largest angle should be opposite the longest side.
Builds from slides 14 to 17, including the final accuracy check.
Why must the formula letter match the opposite side?
Expected elements: mention matching the correct lettered side with the target angle, using the rearranged cosine rule, then checking whether the result makes sense against side lengths and angle size.
What helped you most to choose between sine and cosine rules?
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Which part of solving triangle problems still feels hardest?
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One thing you will remember from today’s lesson
Invite short takeaways such as 'opposite pair', 'included angle', 'cosine first for 3 sides', or 'largest angle opposite longest side'.
Glossary
sine rule - a rule linking each side of a triangle to the sine of its opposite angle
cosine rule - a rule used to find a side from two sides and an included angle, or to find an angle from three sides
included angle - the angle between two known sides
opposite - the side or angle directly across from another part of the triangle
inverse sine - the operation used to find an angle from a sine value
inverse cosine - the operation used to find an angle from a cosine value
triangle sum - the fact that the interior angles of a triangle add to 180°
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