🗺️ Chapter Roadmap
1 The Right-Angled Triangle
Trigonometry is the study of the relationships between the sides and angles of triangles. All basic trig starts with the Right-Angled Triangle (a triangle with one 90° angle).
Hypotenuse:
The longest side, always opposite the 90° angle.
Opposite:
The side directly across from your chosen angle (θ).
Adjacent:
The side next to your angle (θ) that isn't the hypotenuse.
2 SOH-CAH-TOA (The Golden Trick)
There are three main ratios that connect the sides to the angles. Use this famous English phrase to memorize them perfectly:
SOH
Sine = Opposite / Hypotenuse
sin(θ) =
OppHyp
CAH
Cosine = Adjacent / Hypotenuse
cos(θ) =
AdjHyp
TOA
Tangent = Opposite / Adjacent
tan(θ) =
OppAdj
The Reciprocal Ratios (The Flip)
Each ratio has a "partner" which is just the original ratio flipped upside down.
cosec(θ)
1 / sin
sec(θ)
1 / cos
cot(θ)
1 / tan
3 Standard Angles Table
In non-calculator exams (NTS/CSS), you must memorize the values for these five standard angles:
| Ratio | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 12 | 1√2 | √32 | 1 |
| cos | 1 | √32 | 1√2 | 12 | 0 |
| tan | 0 | 1√3 | 1 | √3 | Undefined |
4 Key Identities & Example
Trig Identities
- sin2(θ) + cos2(θ) = 1
- tan(θ) = sin(θ)cos(θ)
📝 Solved Example
A ladder 5m long leans against a wall. It makes a 60° angle with the ground. How high up the wall does it reach?
Angle = 60°
Hypotenuse = 5m
We need "Height" (Opposite).
Use SOH: sin(60°) = Height / 5
(√3 / 2) = Height / 5
Height = (5 × √3) / 2 ≈ 4.33m
Hypotenuse = 5m
We need "Height" (Opposite).
Use SOH: sin(60°) = Height / 5
(√3 / 2) = Height / 5
Height = (5 × √3) / 2 ≈ 4.33m