Step-by-step calculations
● Hypotenuse r
Unit circle → r is always 1
r = 1
● Cosine — horizontal reach
cos θ = adjacent / hypotenuse = x / r
● Sine — vertical reach
sin θ = opposite / hypotenuse = y / r
● Tangent — slope (rise / run)
tan θ = opposite / adjacent = y / x
tan built from sin ÷ cos
sin = y/r, cos = x/r. Divide them — r cancels:
tan θ
=
sin θcos θ
=
(y/r)(x/r)
=
=
Pythagorean identity: sin²θ + cos²θ = 1
what each ratio really means
cos θ — horizontal fraction of r
Travel 1 unit along the hypotenuse →
you cover units horizontally.
The dashed blue arrow on the canvas shows this unit projection.
The dashed blue arrow on the canvas shows this unit projection.
sin θ — vertical fraction of r
Travel 1 unit along the hypotenuse →
you rise units vertically.
The dashed pink arrow on the canvas shows this unit projection.
The dashed pink arrow on the canvas shows this unit projection.
tan θ — rise per unit of run
Move 1 unit horizontally (run) along the x-axis →
the line of the hypotenuse rises units vertically.
The dashed orange column on the canvas shows this. tan = slope of r.
The dashed orange column on the canvas shows this. tan = slope of r.
point P & legend
P = (cos θ, sin θ) =
Hypotenuse r = 1
cos θ (horizontal leg)
sin θ (vertical leg)
tan θ (⊥ to r at P)
1-unit horizontal → cos θ wide
1-unit along r → sin θ tall
1-unit right → tan θ rise