Extending Trigonometric Ratios to Any Angle
Simple Explanation
Right-triangle trigonometry only covers angles between 0° and 90°. To handle any angle, place its vertex at the origin with one ray along the positive x-axis, and take any point (x, y) on the angle's other ray at distance r from the origin. Then define sin θ = y/r, cos θ = x/r, and tan θ = y/x — this works for angles of any size, in any quadrant.
Why Do We Need It?
This single generalization is what makes trigonometry usable beyond right triangles — for rotations, oscillations, and any angle greater than 90° or less than 0°.
See It
A point P at coordinates (-3, 4) in the second quadrant, with a radius line from the origin to P, and dashed lines showing its x and y components
Formula
General Definition of Trigonometric Ratios
sin θ = y/r, cos θ = x/r, tan θ = y/x
For any angle θ (not just acute angles), place its vertex at the origin with one ray along the positive x-axis; take any point (x, y) on the other ray at distance r from the origin, and define the ratios from those coordinates.
- θ
- — the angle, measured counterclockwise from the positive x-axis
- (x, y)
- — any point on the angle's terminal ray
- r
- — the distance from the origin to (x, y); always positive, r = √(x²+y²)
When to use it: Whenever a trigonometric ratio is needed for an angle that is not between 0° and 90°.
Worked Example
Find trigonometric ratios from a point on the terminal ray
A point P(−3, 4) lies on the terminal ray of an angle θ, measured from the origin. Find r, then sin θ, cos θ, and tan θ.
Why Does This Work?
For an acute angle in a right triangle, this definition matches the familiar opposite/hypotenuse and adjacent/hypotenuse ratios exactly — x and y are just signed versions of the triangle's legs, and r is the hypotenuse. Allowing x and y to be negative (depending on the quadrant) is what extends the same idea to angles beyond 90°, without needing a separate definition for each case.
Real-Life Example
Analyzing a rotating Ferris wheel car
An engineer tracks a Ferris wheel car's height as it rotates all the way around, well past the 90° mark.
Right-triangle trigonometry alone cannot describe the car's position past a quarter-turn — the general (x, y, r) definition is required to track it through the full rotation.
Practice
A point P(5, −12) lies on the terminal ray of angle θ. Find r.
MediumCommon mistake
Treating x and y as always positive, the way they are in basic right-triangle trigonometry — in this general definition, x and/or y can be negative depending on the quadrant, which is exactly what makes the ratios correctly negative or positive.
Quick Review
- sin θ = y/r, cos θ = x/r, tan θ = y/x, for any point (x, y) at distance r from the origin.
- r is always positive; x and y can be positive or negative depending on the quadrant.
- This generalizes right-triangle trig to angles of any size.