Trigonometric Ratios of Any Angle

You know that \sin 30° = \tfrac{1}{2}. That comes from a right triangle with a 30° angle — the side opposite to 30° is half the hypotenuse, and there is nothing more to say.

But what is \sin 150°? You cannot draw a right triangle with an interior angle of 150° — no such triangle exists, because the angles of a triangle add to 180°, and a right angle already uses up 90° of that. The right-triangle definition of sine simply does not apply to 150°.

Yet your calculator will happily tell you that \sin 150° = 0.5. The same answer as \sin 30°. That is not a coincidence, and it is not a convention. There is a geometric reason — and to see it, you need a bigger definition of sine that works for every angle.

The unit circle

Take a circle of radius 1, centred at the origin of the coordinate plane. This is called the unit circle. Now pick any angle \theta, measured from the positive x-axis, and rotate counterclockwise by that amount. The ray from the origin at angle \theta hits the unit circle at exactly one point. Call that point P.

The point P has two coordinates. Define:

That is the entire definition. No triangle needed, no restriction on \theta. The angle can be 30°, 150°, -45°, 720°, or \pi/6 radians — the definition always works, because you can always draw a ray at that angle and read off coordinates.

Why this agrees with the right-triangle definition. For an acute angle \theta, drop a perpendicular from P to the x-axis. You get a right triangle with hypotenuse 1 (the radius), adjacent side x_P, and opposite side y_P. The old definitions give \cos\theta = \text{adjacent}/\text{hypotenuse} = x_P/1 = x_P and \sin\theta = \text{opposite}/\text{hypotenuse} = y_P/1 = y_P. The unit circle definition produces the same numbers for acute angles — it is an extension, not a replacement.

Signs in the four quadrants

The unit circle makes the sign question mechanical. In each quadrant, the coordinates of P have definite signs, and those signs determine the signs of every trigonometric ratio.

In Quadrant I, everything is positive. In Quadrant II, only sine is positive (the y-coordinate is positive, the x-coordinate is negative). In Quadrant III, only tangent is positive (both coordinates are negative, and a negative divided by a negative is positive). In Quadrant IV, only cosine is positive.

Indian textbooks often encode this as the mnemonic "All Students Take Coffee" — reading counterclockwise from Quadrant I: All, Sine, Tangent, Cosine. Each word names the ratio(s) that are positive in that quadrant.

Reference angles

Here is the key insight: \sin 150° and \sin 30° have the same absolute value because the two points on the unit circle are mirror images of each other across the y-axis. The point at 150° has coordinates (-\cos 30°, \sin 30°). The sine (the y-coordinate) is the same; the cosine (the x-coordinate) just changes sign.

The acute angle that captures this relationship is called the reference angle.

This is why \sin 150° = \sin 30°. The angle 150° is in Quadrant II, so its reference angle is 180° - 150° = 30°. Sine is positive in Quadrant II. So \sin 150° = +\sin 30° = 1/2.

And \cos 150°? Cosine is negative in Quadrant II. So \cos 150° = -\cos 30° = -\sqrt{3}/2.

The reference angle reduces every problem about arbitrary angles to a problem about acute angles — angles whose values you already know from the standard table (0°, 30°, 45°, 60°, 90°).

Here is the full procedure: given any angle \theta, find its trigonometric ratios in three steps.

1. Find the quadrant that \theta lands in.
2. Find the reference angle \alpha using the table above.
3. Write the ratio of \alpha, and attach the sign from the ASTC rule.

Take \theta = 225°. This is in Quadrant III (between 180° and 270°). The reference angle is 225° - 180° = 45°. In Quadrant III, sine and cosine are both negative, and tangent is positive. So:

Check: \tan 225° = \sin 225° / \cos 225° = (-\sqrt{2}/2) / (-\sqrt{2}/2) = 1. Consistent.

Here is a picture showing the reference angles for all four quadrants, using the same base angle \alpha = 30°:

Allied angles

Angles that differ by multiples of 90° have a systematic relationship. These are called allied angles (or associated angles). The relationships come directly from the unit circle — from the symmetry of a circle about its axes and diagonals.

Here are the key identities. Each one can be read off the unit circle by reflecting or rotating the point P.

Supplementary angles (180° - \theta): the point reflects across the y-axis.

Anti-supplementary angles (180° + \theta): the point reflects through the origin.

Negative angles (-\theta or equivalently 360° - \theta): the point reflects across the x-axis.

Complementary shifts (90° - \theta): this is the one where sine and cosine swap.

The co-function shift (90° + \theta): the point rotates 90° counterclockwise.

There is a pattern hiding here. When the allied angle involves 180° or 360° (even multiples of 90°), the function name stays the same — sine stays sine, cosine stays cosine. When the allied angle involves 90° or 270° (odd multiples of 90°), the function name switches — sine becomes cosine and cosine becomes sine. The sign is then determined by which quadrant the original expression lands in.

Why does the 90° shift swap sine and cosine? On the unit circle, rotating a point by 90° counterclockwise sends (x, y) to (-y, x). The old y-coordinate (sine) becomes the new x-coordinate (cosine), and vice versa. The swap is geometric — it comes from what a 90° rotation does to coordinates.

Proof of \sin(180° - \theta) = \sin\theta. The point at angle \theta has coordinates (\cos\theta, \sin\theta). The point at angle 180° - \theta is the reflection of this point across the y-axis — it has coordinates (-\cos\theta, \sin\theta). The y-coordinate is \sin\theta in both cases. So \sin(180° - \theta) = \sin\theta.

Proof of \cos(180° + \theta) = -\cos\theta. The point at 180° + \theta is diametrically opposite the point at \theta on the unit circle. If \theta gives the point (\cos\theta, \sin\theta), then 180° + \theta gives (-\cos\theta, -\sin\theta). Read off the x-coordinate: \cos(180° + \theta) = -\cos\theta.

Computing ratios: two worked examples

The ratios at the axes

What about angles that land exactly on an axis — 0°, 90°, 180°, 270°? These are not in any quadrant, and they give rise to the boundary values.

At 0°, the point on the unit circle is (1, 0). So \cos 0° = 1 and \sin 0° = 0.

At 90°, the point is (0, 1). So \cos 90° = 0 and \sin 90° = 1. Since \tan 90° = \sin 90° / \cos 90° = 1/0, tangent is undefined at 90°.

At 180°, the point is (-1, 0). So \cos 180° = -1 and \sin 180° = 0.

At 270°, the point is (0, -1). So \cos 270° = 0 and \sin 270° = -1.

These boundary values are worth memorising alongside the standard table. They come up constantly — especially \sin 90° = 1, \cos 90° = 0, and the fact that \tan 90° does not exist.

Common confusions

• "Sine is always positive." It is positive for angles in Quadrants I and II (where the y-coordinate is positive) and negative in Quadrants III and IV. Think of it as the height of the point on the unit circle — below the x-axis means negative height.

• "The reference angle for 120° is 120° - 90° = 30°." The reference angle is measured from the x-axis, not from the y-axis. For 120° (Quadrant II), the reference angle is 180° - 120° = 60°, not 30°.

• "\sin(-\theta) = \sin\theta." Sine is an odd function: \sin(-\theta) = -\sin\theta. It is cosine that is even: \cos(-\theta) = \cos\theta. The unit circle makes this visible — reflecting across the x-axis flips the y-coordinate (sine changes sign) but keeps the x-coordinate (cosine stays).

• "\tan 90° is infinity." It is undefined — not equal to +\infty or -\infty. If you approach 90° from below, \tan\theta \to +\infty. If you approach from above, \tan\theta \to -\infty. Since the two one-sided limits disagree, there is no value at 90° itself.

• "Allied angle formulas must be memorised separately." They do not. All of them follow from the unit circle: decide whether the function name stays or swaps (even vs. odd multiple of 90°), then decide the sign from the quadrant. Two rules, not sixteen formulas.

Going deeper

A general formula for allied angles

Every allied angle has the form n \cdot 90° \pm \theta, where n is an integer. The rule is:

• If n is even, the function name does not change:

  \sin(n \cdot 90° \pm \theta) = \pm \sin\theta, \qquad \cos(n \cdot 90° \pm \theta) = \pm \cos\theta

• If n is odd, the function name switches:

  \sin(n \cdot 90° \pm \theta) = \pm \cos\theta, \qquad \cos(n \cdot 90° \pm \theta) = \mp \sin\theta

In both cases, the sign on the right is determined by checking what sign the original function (the one on the left) would have in the quadrant where n \cdot 90° \pm \theta falls, assuming \theta is acute.

This single rule generates every identity in the allied-angles table — all sixteen entries. It is worth understanding the rule rather than memorising the table, because the rule can be reconstructed from the unit circle in thirty seconds, while the table cannot.

Why 360° is special

Adding 360° to any angle brings the point back to the same position on the unit circle. This means \sin(\theta + 360°) = \sin\theta and \cos(\theta + 360°) = \cos\theta for every \theta. The number 360° (or 2\pi in radians) is the period of sine and cosine.

Tangent has a shorter period: \tan(\theta + 180°) = \tan\theta. This is because the point at \theta + 180° is diametrically opposite the point at \theta, so both coordinates flip sign — and the ratio of two negatives is positive. You will explore this periodicity in detail in Trigonometric Functions and Graphs.

Angles beyond 360° and in radians

The unit-circle definition handles angles of any size without modification. An angle of 750° means rotating counterclockwise by 750° = 2 \times 360° + 30°, which is two full revolutions plus 30°. The terminal point lands at the same position as 30°, so \sin 750° = \sin 30° = 1/2.

In radians, 180° = \pi. All the identities above hold with \pi replacing 180° and \pi/2 replacing 90°. For instance, \sin(\pi - \theta) = \sin\theta. The radian version is what you will use in calculus and in the study of Trigonometric Functions and Graphs.

Where this leads next

The unit circle is the launchpad for everything that follows in trigonometry.

• Trigonometric Functions and Graphs — now that sine and cosine are defined for all angles, you can plot them as functions and study their shape, period, and transformations.
• Trigonometric Identities — the fundamental identities (\sin^2\theta + \cos^2\theta = 1 and its cousins) and strategies for proving new ones.
• Trigonometric Ratios — a refresher on the original right-triangle definitions and standard angle values, if you need one.
• Argument of a Complex Number — the unit circle appears again when complex numbers are written in polar form: the argument of a + bi is exactly the angle \theta whose cosine is a/r and whose sine is b/r.
• Coordinate Geometry Basics — the coordinate framework that makes the unit circle definition possible.