Quadrantal Angles | Moosa Academy

Most angles put their terminal side somewhere inside a quadrant. A few land exactly on an axis: 0°, 90°, 180°, 270° and 360°. These are the quadrantal angles, and their trigonometric values need no memorising — you read them straight off the unit circle.

Concept Angles that land on an axis

A quadrantal angle is an angle in standard position whose terminal side lies along one of the coordinate axes.

On the unit circle the terminal side meets the circle at

(x,\,y)

, and

\cos\theta = x

\sin\theta = y

For a quadrantal angle one coordinate is 0 and the other is ±1.

Reference The complete table of values

θ	Point (x, y)	cos θ	sin θ
0°	(1, 0)	1	0
90°	(0, 1)	0	1
180°	(−1, 0)	−1	0
270°	(0, −1)	0	−1
360°	(1, 0)	1	0

Note Reading the two cases

On the x-axis (0°, 180°, 360°): the height is zero, so $\sin\theta = 0$ , and $\cos\theta = \pm 1$ depending on the direction.

On the y-axis (90°, 270°): the horizontal coordinate is zero, so $\cos\theta = 0$ , and $\sin\theta = \pm 1$ .

Because $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ , the zero cosine at 90° and 270° makes $\tan\theta$ undefined there.

Example Values without a calculator

At 180° the terminal side points along the negative x-axis, meeting the circle at

(-1,\,0)

⟹ cos 180° = −1 and sin 180° = 0

At 270° the point is

(0,\,-1)

⟹ cos 270° = 0 and sin 270° = −1

A full turn brings 360° back to the starting point, so it repeats the values of 0°.

Summary

Quadrantal angles are 0°, 90°, 180°, 270° and 360° — their terminal side lies on an axis.
On the unit circle cos θ = x and sin θ = y, and for these angles one coordinate is always 0.
On the x-axis: sin = 0 and cos = ±1. On the y-axis: cos = 0 and sin = ±1.
0° and 360° share the same position, so they share the same values.