SumNinja

Why tangent graphs have gaps

About 4 min

Tangent compares how far up and across a direction goes. Divide the height by the left-right value. A straight-up aim has no width, so its graph has a gap.

Imagine aiming a pointer in a game. The circle below measures one unit from centre to edge. The tip's height is sine; its left-right value is cosine. Tangent divides height by that left-right value. Turn the pointer to see why aiming straight up causes trouble.

Coordinates on a unit circle Radius one. Right: 0 degrees, coordinates (1, 0). Top: 90 degrees or pi over two, coordinates (0, 1). Left: 180 degrees or pi, coordinates (-1, 0). Bottom: 270 degrees or three pi over two, coordinates (0, -1). Cosine is the horizontal coordinate and sine is vertical. 90° = π/2 · (0, 1)270° = 3π/2 · (0, −1)180° = π(−1, 0)0° = 0(1, 0)
Blue tracks left and right: cosine. Green tracks height: sine. The radius is one. A full turn is 360°, or 2π radians.

cos ≈ 0.707; sin ≈ 0.707; tan = 1

Follow the pointer through three directions

Read tangent as height ÷ sideways position.

  1. Aim level to the right

    tan⁡(0)=sin⁡(0)cos⁡(0)=01=0\tan(0)=\frac{\sin(0)}{\cos(0)}=\frac01=0

    The tip sits one unit right of the centre, with no height. Zero divided by one gives zero.

  2. Give height and width the same size

    tan⁡(45∘)=tan⁡(π/4)=1\tan(45^\circ)=\tan(\pi/4)=1

    At 45 degrees, the two lengths match and both are positive. Dividing equal lengths gives one.

  3. Aim straight up

    tan⁡(90∘)=10undefined\tan(90^\circ)=\frac10\quad\text{undefined}

    At the top, sideways position is zero. No number times zero can give a height of one, so this division has no answer.

A vertical aim makes a break

Each side of a gap belongs to its own piece of the curve. Never draw a line across the missing value.

Connect the aim to the graph

The graph puts angle along the bottom and tangent up the side. A level aim gives (0, 0). At 45°, or π/4 radians, the value is one.

As the pointer nears straight up, its height stays near one while its sideways length shrinks. Dividing by that tiny length makes the result large.

Just past straight up, the sideways position turns negative while the height stays positive. The result turns negative. The graph has separate pieces with a gap between them. The gap itself is not infinity.

Read the angle labels

Degrees and radians both measure turns. A full turn is 360°, or 2π radians. The symbol π is about 3.14. A quarter turn is 90°, or π/2 radians. The game uses radians.

Half a turn reverses both signs, which keeps the ratio the same. So tangent repeats every 180°, or π radians. Its gaps are at π/2, then 3π/2, and so on.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Turn below the horizontal

tan⁡(−π/4)\tan(-\pi/4)

Aim one eighth of a turn below the right-hand direction. What value does tangent have there?

Show the answerHide the answer
−1-1

Sine is negative and cosine is positive. Their sizes match, so the answer is −1.

Identify an undefined value

tan⁡(π/2)\tan(\pi/2)

What happens to tangent at this angle? Explain why it cannot be zero.

Show the answerHide the answer
undefined\text{undefined}

No number times zero gives one, so 1 ÷ 0 has no answer. It is neither zero nor infinity.

Pick a graph that can break

y=tan⁡(x)y=\tan(x)

Both pass through zero. Which can represent a vertical aim with no defined ratio?

Graph A

Separate pieces have gaps at a quarter turn and then every half turn.

Graph B

One smooth wave stays between minus one and one without gaps.

Show the answerHide the answer
Graph A\text{Graph A}

Graph A has gaps where the left-right value is zero. Graph B stays between minus one and one with no gaps, so it cannot be tangent.

When the game stretches the graph

In y = 2tan(x), double every height on the graph. The gaps stay put. Adding a number moves the graph up or down. A minus sign flips the heights but leaves the gap angles the same.

In y = tan(2x), the pointer turns twice as fast. The graph repeats after π/2 instead of π. Its gaps move closer together too. Check the full rule before you choose.

Put the gap idea to the test

Match tangent rules to graphs. Look for a known value, the gaps and the distance before the shape repeats. The game uses radians.

2-minute practice · keep going after mistakes · opens in a new tab

Try tangent graph questions