Law of Sines & Cosines explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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The law of sines and law of cosines help solve triangles that are not righngled.
They connect side lengths and angles in any triangle.
They are used in surveying, navigation, physics, geometry, and advanced trigonometry.
Righriangle ratios work only when there is a right angle.
For any triangle, the sine rule relates each side to the sine of its opposite angle.
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The cosine rule relates one side to the other two sides and the included angle.
Use the sine rule when you know an opposite sidngle pair.
Use the cosine rule when you know two sides and the included angle, or all three sides.
Meaning: A side and the angle across from it form an opposite pair.
Definition: Side a is opposite angle A.
Matching sides and opposite angles correctly is essential.
Meaning: The sine rule compares sides with sines of opposite angles.
Definition: in in in C.
It works best when one opposite pair is known.
Meaning: The cosine rule is a noighriangle version of Pythagoras with an angle term.
Definition: bc cos A.
Use it for SAS or SSS triangle problems.
Meaning: The given information decides the rule.
Definition: Use sine rule for known opposite pair. Use cosine rule for SAS or SSS.
A quick labenheck step prevents wrong rule choice.
Easy
Question: Which side is opposite angle A?
Medium
Question: In a triangle, , degrees, and degrees. Find b.
Exam-level
Question: Use the cosine rule to find side a when , , and degrees.
Pairing a side with the wrong angle
Why it happens: The triangle is not labelled.
Correct approach: Label opposite pairs before using a formula.
Using sine rule without an opposite pair
Why it happens: The given information is not checked.
Correct approach: Use cosine rule when there is no known opposite pair.
Using the wrong included angle in cosine rule
Why it happens: The angle between two known sides is missed.
Correct approach: A must be between sides b and c in formula.
Can these rules be used in any triangle?
Yes. They work for noight and right triangles.
When should I use cosine rule?
Use it for two sides with included angle, or when all three sides are known.
What is an opposite pair?
It is a side and the angle directly across from it.
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