Trigonometric Ratios of Quadrantal Angles
Simple Explanation
At 0°, 90°, 180°, 270°, and 360°, the terminal ray lies exactly ON an axis rather than inside a quadrant, giving simple exact values: sin and cos are always 0, 1, or −1 at these angles, and tan is undefined whenever the point lies on the y-axis (at 90° and 270°), since that means x=0.
Why Do We Need It?
These exact values come up constantly as starting/ending points of a full rotation, and knowing where tan is undefined avoids a division-by-zero error.
See It
A unit circle with four points marked at 0 degrees (1,0), 90 degrees (0,1), 180 degrees (-1,0), and 270 degrees (0,-1)
Formula
Trigonometric Ratios of Quadrantal Angles
sin: 0, 1, 0, −1, 0. cos: 1, 0, −1, 0, 1. tan: 0, undefined, 0, undefined, 0 — at 0°, 90°, 180°, 270°, 360°
The sine, cosine, and tangent values at the four axis-aligned "quadrantal" angles (and back to 0°/360°), found by taking a point directly on an axis rather than inside a quadrant.
- 0°, 360°
- — point on the positive x-axis: (x,y) = (r, 0)
- 90°
- — point on the positive y-axis: (x,y) = (0, r)
- 180°
- — point on the negative x-axis: (x,y) = (−r, 0)
- 270°
- — point on the negative y-axis: (x,y) = (0, −r)
When to use it: Whenever a trigonometric ratio is needed for exactly 0°, 90°, 180°, 270°, or 360° — including recognizing where tan is undefined.
Worked Example
Evaluate tan at two quadrantal angles
Evaluate tan 180°, and explain why tan 90° is undefined.
Why Does This Work?
These values fall directly out of the general definition sin θ=y/r, cos θ=x/r, tan θ=y/x, applied to the specific points where the terminal ray meets an axis exactly — no separate rule is needed beyond the definition itself.
Real-Life Example
Identifying the extremes of a pendulum's swing
An oscillating pendulum's angle passes through 0°, 90°, 180°, and 270° as it swings through a full cycle.
The quadrantal values instantly identify exact zero-crossing and peak points of the motion, without needing a calculator.
Practice
Find sin 270°.
MediumCommon mistake
Assuming tan is always defined for every angle — tan θ is undefined whenever cos θ=0, which happens exactly at 90° and 270°.
Quick Review
- sin: 0, 1, 0, −1, 0 at 0°, 90°, 180°, 270°, 360°.
- cos: 1, 0, −1, 0, 1 at the same five angles.
- tan is undefined at 90° and 270°, where x=0.