Trigonometric Functions
“Trig functions are the height and width of a point circling around.”
The formula
sin²θ + cos²θ = 1How to read it: on the unit circle, a point's vertical coordinate is sinθ and its horizontal is cosθ, and their squares always add to 1
- sinθ
- — the vertical coordinate (height) of the point on the unit circle
- cosθ
- — the horizontal coordinate (base) of the point on the unit circle
- tanθ
- — sinθ/cosθ — the slope of the line to that point
The hook
Trig functions are really 'the shadow of a point going around a circle' — feed in an angle and out pop a height (sin) and a width (cos).
In plain words
On a circle of radius 1, after turning by angle θ, the point's vertical position is sinθ and its horizontal position is cosθ. tanθ is their ratio.
The intuition
Spin a radius-1 ruler about the origin by angle θ. The tip's height is sin, how far it went sideways is cos. As θ grows, the point circles around and both values wave up and down between −1 and 1 — which is why trig functions become waves.
How it's built
The point's coordinates are (cosθ, sinθ). It sits on the radius-1 circle, so by Pythagoras (horizontal)²+(vertical)² = 1², i.e. sin²θ+cos²θ=1. Every trig identity flows from this one circle.
Example
At θ=30°, sin30°=1/2 and cos30°=√3/2. Check: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1. The identity holds exactly.
Common mistake
sin²θ means (sinθ)², not sin(θ²). And sin(A+B) ≠ sinA + sinB — you cannot just distribute over the angle inside.
Where it's used
Waves of sound, light, and current; the swing of pendulums and springs; circular motion; measuring distance by triangulation — the math of everything that spins and oscillates.
Where it came from
'Sine' traces back to the Sanskrit word for a bowstring (jya), mistranslated through Arabic into the Latin sinus ('bay, curve'), and the name stuck.
Prerequisites
Quick check
What is sin 30°?
- 1/2✓
- √3/2
- 1
- 0
Practice
Find sin 90°.
Answer: 1
- 90° is the top of the circle, point (0,1)
- The vertical coordinate is 1 → sin 90° = 1
Takeaway: Sin is the 'height' of the point on the circle.
Find cos 60°.
Answer: 0.5
- The horizontal coordinate at 60°
- cos 60° = 1/2 = 0.5
Takeaway: Cos is the 'width' of the point on the circle.
Find tan 45°.
Answer: 1
- tan 45° = sin45°/cos45° = (√2/2)/(√2/2)
- = 1
Takeaway: At 45° height equals width, so the slope is 1.
Verify by calculation that sin²θ + cos²θ = 1 holds at θ=30°.
Answer: undefined
- sin30°=1/2, cos30°=√3/2
- (1/2)² + (√3/2)² = 1/4 + 3/4
- = 1
Takeaway: This identity is Pythagoras on the radius-1 circle.