LLL - Trigonometry
Pythagoras and Trig Foundations
Build Pythagoras from side identification to square-root choice, then move into trig ratios from a marked angle.
Right triangles first. Ratios second.
This lesson builds the chain students need before calculation: short sides, long side, square root from triples, missing sides, then SOH CAH TOA.
Pythagoras
short^2 + short^2 = long^2
SOH CAH TOA
The marked angle decides which side is opposite and which side is adjacent.
Teacher refresher
Pythagoras: short^2 + short^2 = long^2
The long side is opposite the right angle. The two sides that make the right angle are the short sides.
Long side
Two short sides
Pythagoras rule
short^2 + short^2 = long^2
When the long side is missing, add the two short-side squares. When a short side is missing, subtract from the long-side square.
Missing long side
3^2 + 4^2 = long^2
Missing short side
5^2 - 3^2 = short^2
Side names from an angle
Opposite, adjacent, hypotenuse
The hypotenuse is fixed by the right angle. Opposite and adjacent change when the marked angle changes.
From the marked angle
The key test
Opposite is across from the marked angle.
Adjacent touches the marked angle and is not the hypotenuse.
Hypotenuse is opposite the right angle.
SOH CAH TOA
Choose the ratio from the side pair
When the words say which two sides are involved, the ratio is fixed.
SOH
sin = opposite / hypotenuse
CAH
cos = adjacent / hypotenuse
TOA
tan = opposite / adjacent
Word-pair drill
The next section shows just the words first. Students choose the trig ratio before any calculation.
Numbered triangles
Use the ratio that connects x and the given side
Students are not calculating yet. They are choosing the most helpful ratio from the labelled triangle.
Look at the two sides
Decision path
Question
Answer this page before moving on.
Answer
Use the diagram.
Complete
Foundation chain complete.
You have identified Pythagoras sides, drilled square-root choices, solved mixed Pythagoras questions, named trig sides, and chosen useful trig ratios.