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Quadrantal Angle

Quadrantal Angle

An angle with terminal side on the x-axis or y-axis. That is, the angles 0°, 90°, 180°, 270°, 360°, 450°, ... as well as –90°, –180°, –270°, –360°, ...

 

 

See also

Acute angle, obtuse angle

Key Formula

θ=n⋅90∘where n is any integer\theta = n \cdot 90^\circ \quad \text{where } n \text{ is any integer}
Where:
  • θ\theta = The quadrantal angle
  • nn = Any integer (…, −2, −1, 0, 1, 2, 3, …)

Worked Example

Problem: Determine the sine, cosine, and tangent of the quadrantal angle 270°.
Step 1: Place the angle in standard position. The initial side lies along the positive x-axis. Rotating 270° counterclockwise lands the terminal side on the negative y-axis.
Step 2: Identify the point where the terminal side intersects the unit circle. At 270°, this point is (0, −1).
(x,y)=(0,−1)(x, y) = (0, -1)
Step 3: Read off the trig values using the unit circle definitions: cosine equals the x-coordinate, sine equals the y-coordinate.
cos⁡270°=0,sin⁡270°=−1\cos 270° = 0, \quad \sin 270° = -1
Step 4: Compute tangent as sine divided by cosine.
tan⁡270°=sin⁡270°cos⁡270°=−10=undefined\tan 270° = \frac{\sin 270°}{\cos 270°} = \frac{-1}{0} = \text{undefined}
Answer: For 270°: sin = −1, cos = 0, and tan is undefined because division by zero occurs.

Another Example

Problem: List all quadrantal angles between −360° and 360° (inclusive) and state which axis the terminal side falls on.
Step 1: Use the formula θ = n · 90° and find every integer n that gives an angle in the range [−360°, 360°].
n=−4,−3,−2,−1,0,1,2,3,4n = -4, -3, -2, -1, 0, 1, 2, 3, 4
Step 2: Compute each angle and identify the axis. Angles that are multiples of 180° land on the x-axis; odd multiples of 90° land on the y-axis.
Step 3: x-axis: −360°, −180°, 0°, 180°, 360°. y-axis: −270°, −90°, 90°, 270°.
Answer: There are nine quadrantal angles in that range: −360°, −270°, −180°, −90°, 0°, 90°, 180°, 270°, and 360°. Five have terminal sides on the x-axis and four on the y-axis.

Frequently Asked Questions

Why is it called a quadrantal angle?
The name comes from the word 'quadrant.' The four quadrants of the coordinate plane are separated by the x-axis and y-axis. A quadrantal angle's terminal side falls exactly on one of these boundary axes, so it sits between quadrants rather than inside any single one.
What are the trig values of all quadrantal angles?
At 0°: sin = 0, cos = 1, tan = 0. At 90°: sin = 1, cos = 0, tan is undefined. At 180°: sin = 0, cos = −1, tan = 0. At 270°: sin = −1, cos = 0, tan is undefined. Any coterminal quadrantal angle shares the same values as the corresponding angle in this list.

Quadrantal Angle vs. Reference Angle

A quadrantal angle has its terminal side on an axis (0°, 90°, 180°, 270°). A reference angle is the acute angle between any angle's terminal side and the nearest part of the x-axis. Quadrantal angles have a reference angle of either 0° or 90°, which are not themselves acute, so many textbooks say the reference angle is not defined for quadrantal angles.

Why It Matters

Quadrantal angles are boundary cases that appear constantly in trigonometry. At these angles, sine or cosine equals zero, which makes certain trig functions (tangent, cotangent, secant, or cosecant) undefined. Recognizing a quadrantal angle quickly tells you whether a trig expression is 0, ±1, or undefined—saving time on tests and preventing division-by-zero errors in calculations.

Common Mistakes

Mistake: Thinking tan 90° or tan 270° equals zero instead of being undefined.
Correction: At 90° and 270°, cosine is 0, so tan = sin/cos involves division by zero. Tangent is undefined at these angles, not zero. Tangent equals zero at 0° and 180°, where sine is 0.
Mistake: Assuming quadrantal angles belong to a specific quadrant.
Correction: A quadrantal angle's terminal side lies on an axis, which is the boundary between quadrants. It does not belong to any single quadrant. This matters when applying sign rules for trig functions.

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