Introduction
Feed any angle into a calculator and \(\sin\) and \(\cos\) always return a value between \(-1\) and \(1\). \(\sin 89° = 0.9998\), \(\cos 89° = 0.0175\) — no matter how large the angle, they never breach that narrow fence. Yet feed the same \(89°\) into \(\tan\) and suddenly you get \(57.29\), a large number. At \(89.9°\) it is \(572.96\), at \(89.99°\) it is \(5729.6\)…, and at exactly \(90°\) the calculator throws an error.
All three are trigonometric functions, so why do sine and cosine sit tamely between \(-1\) and \(1\) while tangent alone shoots off to infinity? The answer is that the three are different kinds of thing. Sine and cosine are 'coordinates'; tangent is a 'ratio of coordinates'. That one small distinction decides the fate of the values.
One piece of terminology first. When we measure an angle, the ray with one end fixed at the origin and rotated by the given angle is called the terminal side. Like the second hand of a clock, a needle starting from an axis and turned by an angle \(\theta\) (theta, the Greek letter for an angle) is the terminal side.
1. Why sine and cosine are trapped between −1 and 1 — because they are 'coordinates'
Let us first view the trig functions on a circle. On a circle of radius \(1\) (the unit circle), the terminal side turns by an angle \(\theta\) and meets the circle at a point \(P\); that point's horizontal coordinate is the cosine, its vertical coordinate the sine.
$$\cos\theta = (\text{the horizontal coordinate of } P),\qquad \sin\theta = (\text{the vertical coordinate of } P)$$The reason the values are trapped is now immediately visible. The point \(P\) lies on a circle of radius \(1\), so however far it gets from the origin, its distance is \(1\). Hence its horizontal coordinate and its vertical coordinate can neither drop below \(-1\) nor rise above \(1\).
$$-1 \le \cos\theta \le 1,\qquad -1 \le \sin\theta \le 1$$That sine and cosine live inside a narrow fence is not a rule to memorize or a coincidence — it is because they are the coordinates of a point on a circle of radius \(1\). The point cannot leave the circle, so the coordinates cannot leave \(1\). (Redefining the trigonometric ratios as coordinates on a circle like this is covered in detail in A Right Triangle Only Reaches 90° — So How Do We Measure sin 120°?.)
2. Tangent is not a coordinate but a 'ratio of coordinates'
Tangent is a different matter. Tangent is not one of the point \(P\)'s coordinates but the vertical coordinate divided by the horizontal coordinate.
$$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\text{vertical coordinate}}{\text{horizontal coordinate}} = \frac{y}{x}$$In words: sine and cosine read off 'where the point is' directly, while tangent measures 'how steep a direction the point lies in, seen from the origin'. In fact \(y/x\) is exactly the slope of the line (the terminal side) joining the origin to \(P\) — go across by \(x\) and you rise by \(y\).
And a 'ratio' has no reason to be trapped. Each coordinate alone cannot exceed \(1\), but dividing a small number by an even smaller one makes the quotient as large as you like. Hold the numerator near \(1\) and shrink only the denominator, \(0.5 \to 0.1 \to 0.0175\), and the quotient leaps \(2 \to 10 \to 57\). The numerator stays put, yet the shrinking denominator makes the quotient explode. This — dividing — is exactly where tangent's fate parts from that of the fenced-in sine and cosine.
3. What happens near 90° — divide by something heading to zero and it explodes
So why blow up at \(90°\) in particular? As the angle approaches \(90°\), the point \(P\) climbs to the top of the circle, \((0,\,1)\). There the vertical coordinate (sine) approaches \(1\), but the horizontal coordinate (cosine) approaches \(0\). Since tangent divides by this horizontal coordinate, the closer the divisor gets to \(0\), the more endlessly the quotient grows.
$$\tan 89° = \frac{0.9998}{0.0175} \approx 57.29,\qquad \tan 89.9° \approx 572.96$$The more tightly you press the angle toward \(90°\), the smaller the denominator and the higher tangent shoots. And at exactly \(90°\) the horizontal coordinate is precisely \(0\). Dividing by \(0\) is not allowed in mathematics, so \(\tan 90°\) has no value at all. That is why the calculator errors out. On the graph a vertical boundary line appears here that the function can never touch, and this is called an asymptote — a line the curve approaches endlessly but never meets.
Look a little more closely at direction and there is one more curious thing. Approach \(90°\) from the left (say \(89.9°\)) and the horizontal coordinate is a tiny positive number, so tangent shoots up toward \(+\infty\); but just past \(90°\) (say \(90.1°\)) the point \(P\) crosses to the left and the horizontal coordinate becomes a tiny negative number. So at \(90°\) tangent jumps in an instant from \(+\infty\) to \(-\infty\). It is a dramatic scene that sine and cosine never stage — belonging only to tangent, which carries a division inside it.
One more note: the very name 'tangent' is tied to this story. Stand a vertical tangent line just touching the circle at its right edge \((1,\,0)\), extend the terminal side until it meets that line, and the height of the meeting point is exactly \(\tan\theta\). As the angle approaches \(90°\) and the terminal side stands vertical, that meeting point flees all the way to the sky. You can watch this scene for yourself in the interactive below.
4. See it for yourself
In the interactive below, sweep the \(\theta\) slider from \(0°\) up toward \(90°\). On the left circle the vertical coordinate (sine) and horizontal coordinate (cosine) of the point \(P\) are shown, and if you extend the terminal side to meet the vertical tangent line on the right (the line just touching the circle), the height of that meeting point is the tangent value. The closer \(\theta\) gets to \(90°\), the further this point flees upward.
The bar comparison below places all three values side by side on one scale. Grow the angle and the sine and cosine bars only rise and fall inside the gray band between \(-1\) and \(1\), but the tangent bar — the ratio of the two coordinates — at some point bursts through the band and shoots off the top of the screen. Use the 'example angle' buttons to press \(45°\) (tangent \(1\)), \(60°\) (about \(1.73\)), \(80°\) (about \(5.67\)), and \(89°\) (about \(57.29\)) and watch how the value explodes.
Key takeaways
- Sine and cosine are the coordinates of a point on a circle of radius \(1\). The point cannot leave the circle, so the coordinates stay trapped between \(-1\) and \(1\).
- Tangent is not a coordinate but the ratio of the two coordinates (vertical ÷ horizontal = \(y/x\)), which is also the slope of the terminal side. A ratio has no ceiling, so as the denominator shrinks it can grow as large as you like.
- As the angle approaches \(90°\), the denominator — the horizontal coordinate (cosine) — approaches \(0\), so tangent shoots to infinity, and at exactly \(90°\) you divide by \(0\) and there is no value (an asymptote). Passing through \(90°\), the sign flips from \(+\) to \(-\).
Whether a value stays trapped or shoots up came down, in the end, to what the function 'measures'. Measure a coordinate and you stay inside the circle; measure a ratio of coordinates and the moment the denominator vanishes it opens to the sky. This single difference splits the whole character of sine and cosine from that of tangent.
Related reading — A Right Triangle Only Reaches 90° — So How Do We Measure sin 120°? (redefining the ratios as coordinates on a circle — this post's starting point) · Why Are the Trig Ratios the Same Regardless of Triangle Size? (why sine, cosine, and tangent are 'ratios', and their three reciprocals) · Why Do −60° and 300° Have the Same Sine? (trig functions seen as direction — periodicity and negative angles)