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Pythagoras’ Theorem
What is it?
Pythagoras’ Theorem helps you find the length of a side in a right-angled triangle. It’s like a magic rule that connects all three sides.
The Rule
If a triangle has a right angle (90°), then:
a² + b² = c²
a and b are the shorter sides (called "legs")
c is the longest side (called the "hypotenuse") — it’s always opposite the right angle
Trigonometry in Right-Angled Triangles
What’s the Big Idea?
Trigonometry helps you find missing angles or sides in a right-angled triangle using special ratios.
The Three Magic Ratios - use SOH CAH TOA to remember them:
θ is the angle you're working with
Opposite = side across from the angle
Adjacent = side next to the angle (not the hypotenuse)
The perpendicular distance from a point to a line is the shortest distance to the line
Explained:
These values are exact, not rounded.
You’ll often use them in non-calculator papers.
They come up in geometry, trigonometry, and coordinate problems.
Memorizing this table saves time and builds confidence
To solve things like sin(x) = 0.5, get your calculator and press shift sin then 0.5 = sin⁻¹(0.5)
Symmetry Rules
sin(x) is positive in Quadrants I and II (since it's y)
cos(x) is positive in Quadrants I and IV (since it's x)
tan(x) is positive in Quadrants I and III (since it's y/x so both need to be positive/negative)
Remember by the mnemonic : All Students Take Calculus (Quad 1, Quad 2, 3, and 4 consecutively)
Use these rules to find secondary solutions:
If sin(x) = 0.5 → x = 30° and 150°
If cos(x) = 0.5 → x = 60° and 300°
If tan(x) = 1 → x = 45° and 225°
Sine and Cosine Rules:
Sine Rules:
Use when you know:
Two angles and one opposite side (AAS or ASA)
Two sides and one opposite angle (SSA)
Cosine Rule:
c² = a² + b² – [2ab × cos(C)]
Use when you know:
All three sides (SSS)
Two sides and the included angle (SAS)
Non-right Angled Triangle Area:
Area = ½ × a × b × sin(C)
Use when you know:
Two sides and the angle between them (SAS)
This formula is not for right-angled triangles—use it for general triangles
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