Sign of Trigonometric Functions

When you evaluate a trigonometric function at an angle, the result can be positive or negative depending on which quadrant of the coordinate plane the terminal side of the angle falls in. Understanding why certain functions are positive in some quadrants and negative in others is not just a memorization trick — it follows directly from the definitions of sine, cosine, and tangent in terms of x- and y-coordinates on the unit circle. Recall that for a point P = (x, y) on the unit circle (a circle of radius 1 centered at the origin), the cosine of the angle θ is defined as the x-coordinate of P, the sine is the y-coordinate, and the tangent is the ratio y/x. Because x and y can each be positive or negative depending on the quadrant, the signs of these ratios change in a predictable pattern across the four quadrants. Mastering the sign pattern of trigonometric functions is essential for solving equations, simplifying expressions using reference angles, and working with inverse trigonometric functions. The mnemonic 'All Students Take Calculus' (ASTC) is a popular memory aid, but the real goal is to understand the geometry behind the pattern so you can reconstruct it at any time without memorization.

Key takeaways
  • The Unit Circle Definitions
  • Quadrant I (0° < θ < 90°)
  • Quadrant II (90° < θ < 180°)
  • Quadrant III (180° < θ < 270°)
  • Quadrant IV (270° < θ < 360°)
  • The ASTC Mnemonic
  • Using Reference Angles

The Unit Circle Definitions

Place an angle θ in standard position (vertex at the origin, initial side along the positive x-axis). The terminal side intersects the unit circle at a point (x, y). By definition: sin θ = y, cos θ = x, and tan θ = y/x (provided x ≠ 0). These coordinate values determine all signs.

Quadrant I (0° < θ < 90°)

In Quadrant I both x > 0 and y > 0. Therefore sin θ > 0, cos θ > 0, and tan θ = y/x > 0. ALL three primary functions are positive. For example, sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 — all positive.

Quadrant II (90° < θ < 180°)

In Quadrant II x < 0 and y > 0. Therefore sin θ > 0 (y is positive), cos θ < 0 (x is negative), and tan θ = y/x < 0 (positive divided by negative). Only SINE is positive. For example, sin 150° = sin 30° = 1/2 > 0, but cos 150° = −√3/2 < 0 and tan 150° = −1/√3 < 0.

Quadrant III (180° < θ < 270°)

In Quadrant III both x < 0 and y < 0. Therefore sin θ < 0, cos θ < 0, but tan θ = y/x = (negative)/(negative) > 0. Only TANGENT is positive. For example, sin 210° = −1/2, cos 210° = −√3/2, but tan 210° = (−1/2)/(−√3/2) = 1/√3 > 0.

Quadrant IV (270° < θ < 360°)

In Quadrant IV x > 0 and y < 0. Therefore sin θ < 0 (y negative), cos θ > 0 (x positive), and tan θ = y/x < 0 (negative divided by positive). Only COSINE is positive. For example, sin 330° = −1/2, cos 330° = √3/2, tan 330° = −1/√3.

The ASTC Mnemonic

The phrase 'All Students Take Calculus' gives the positive function(s) starting from Quadrant I going counterclockwise: A = All (QI), S = Sine (QII), T = Tangent (QIII), C = Cosine (QIV). Reciprocal functions follow the same sign as their reciprocal partner: csc θ has the same sign as sin θ, sec θ the same sign as cos θ, and cot θ the same sign as tan θ.

Using Reference Angles

The reference angle θ' is the acute angle between the terminal side and the x-axis. The magnitude of any trig function equals the value of that function at the reference angle; you then apply the sign from the quadrant rule. For instance, to find sin 210°: the reference angle is 210° − 180° = 30°, sin 30° = 1/2, and in QIII sine is negative, so sin 210° = −1/2.

The Key Equation

sin⁡θ=y,cos⁡θ=x,tan⁡θ=yx\sin\theta = y,\quad \cos\theta = x,\quad \tan\theta = \frac{y}{x}
  • \theta — Angle (degrees or radians): The angle in standard position measured counterclockwise from the positive x-axis to the terminal side.
  • x — x-coordinate (dimensionless): The horizontal coordinate of the point where the terminal side meets the unit circle; equals cos θ. Positive in QI and QIV, negative in QII and QIII.
  • y — y-coordinate (dimensionless): The vertical coordinate of the point where the terminal side meets the unit circle; equals sin θ. Positive in QI and QII, negative in QIII and QIV.
  • \frac{y}{x} — Tangent ratio (dimensionless): The ratio of the y-coordinate to the x-coordinate. Positive when x and y have the same sign (QI and QIII), negative when they differ (QII and QIV).

Worked Example

Problem. Determine the sign of each of the six trigonometric functions for the angle θ = 315°. Then state the exact value of sin 315°, cos 315°, and tan 315°.

sin⁡315∘=−22,cos⁡315∘=22,tan⁡315∘=−1\sin 315^\circ = -\dfrac{\sqrt{2}}{2},\quad \cos 315^\circ = \dfrac{\sqrt{2}}{2},\quad \tan 315^\circ = -1

Answer. sin 315° = −√2/2, cos 315° = √2/2, tan 315° = −1. Sine and tangent are negative; cosine is positive.

Figure
Diagram
xyQuadrant Isin +cos +tan +(ALL)Quadrant IIsin +cos −tan −(SINE)Quadrant IIIsin −cos −tan +(TANGENT)Quadrant IVsin −cos +tan −(COSINE)O
Origins · 1 minHipparchus of Nicaea and how this was discovered
HO
Hipparchus of Nicaea
-190–-120 · Greek

Hipparchus (c. 190–120 BCE) is widely regarded as the founder of trigonometry. He constructed the first known trigonometric table, a table of chords (closely related to the sine function), which he used to solve problems in spherical astronomy. His systematic treatment of the relationship between angles and lengths on a circle laid the groundwork for all later developments in trigonometry, including the unit-circle approach that underpins our understanding of sign conventions today. Hipparchus discovered the precession of the equinoxes — the slow wobble of Earth's rotational axis over a ~26,000-year cycle — entirely through careful naked-eye astronomical observations, without a telescope.

  • The Unit Circle — The sign rules for trig functions follow directly from x and y coordinates on the unit circle, making this the foundational concept for understanding sign changes.
  • Reference Angles — Reference angles allow you to find the magnitude of any trig function; the quadrant sign rules then determine the correct sign of the result.
  • Trigonometric Identities — Many identities such as sin(π − θ) = sin θ and cos(π − θ) = −cos θ are direct consequences of the sign pattern across quadrants.
  • Inverse Trigonometric Functions — Knowing which quadrant an angle lies in is essential when using inverse trig functions, since calculators only return principal values and you must adjust for the correct quadrant.