SumNinja

Understand sine and cosine with one circle

About 4 min

Sine follows a point’s height around a circle. Cosine follows its left-right value. Plot those values through a turn to make two waves.

A game light sweeps in a circle. You want to track its height and tell the two wave graphs apart. The circle below measures one unit from centre to edge. Its tip’s height is sine, and its left-right value is cosine. Turning the pointer changes both at once.

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; sin = 1

0π/2π3π/22π1−1

Sine: solid green. Cosine: dashed blue. Read the angle along the bottom and the circle’s value up the side.

Follow the same point through a turn

Read the pair as (cosine, sine): left-right first, then height.

  1. Start at the far right

    0∘:(cos⁡,sin⁡)=(1,0)0^\circ:\quad(\cos,\sin)=(1,0)

    The tip is one unit right of the centre and has no height. Start here, then turn anticlockwise.

  2. Turn one quarter to the top

    90∘:(cos⁡,sin⁡)=(0,1)90^\circ:\quad(\cos,\sin)=(0,1)

    The sideways value reaches zero while the height reaches one. The two wave markers show these values at the same angle.

  3. Turn another quarter to the left

    180∘:(cos⁡,sin⁡)=(−1,0)180^\circ:\quad(\cos,\sin)=(-1,0)

    Left of the centre means a negative left-right value. The height is back at zero. At the bottom, the height becomes minus one.

See why the waves start differently

The wave graphs put angle along the bottom and the circle’s value up the side. At zero, the green sine wave starts at zero; the dashed blue cosine wave starts at one.

At a quarter turn, sine reaches one while cosine reaches zero. Use the control to match the circle’s two values with the two markers. A full turn brings the point and both waves back to their starts.

The circle has radius one: its centre-to-edge distance is one. So neither height nor left-right value can go beyond one or minus one. Both waves stay between those bounds.

Read the same turn in degrees or radians

Degrees and radians are two angle units. A full turn is 360°, or 2π radians. The symbol π is about 3.14. A half turn is π radians, and a quarter turn is π/2.

The graph game uses radians. Read its angle labels before you compare a height. The value on the wave is a position from the circle, not another angle.

Try it yourself

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

Read the left-hand point

cos⁡(π)=?sin⁡(π)=?\cos(\pi)=?\qquad\sin(\pi)=?

Which value gives height and which gives left-right position?

Show the answerHide the answer
cos⁡(π)=−1sin⁡(π)=0\begin{gathered}\cos(\pi)=-1\\\sin(\pi)=0\end{gathered}

Cosine gives the left-right value, −1. Sine gives the height, zero.

Read the bottom point

sin⁡(3π/2)=?\sin(3\pi/2)=?

What height does the light reach at the bottom of its circle?

Show the answerHide the answer
−1-1

The point sits one unit below the centre, so its sine is −1.

Choose cosine from two waves

y=cos⁡(x)y=\cos(x)

Check where the light starts at angle zero, then compare the quarter turn.

Graph A

At zero the value is zero. At a quarter turn it is one.

Graph B

At zero the value is one. At a quarter turn it is zero.

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

Cosine starts at one and reaches zero after a quarter turn. Graph B fits. Graph A follows height instead, so it is sine.

Change a wave’s height or speed

In y = 2sin(x), double every height. The wave runs from minus two to two and still repeats after 2π. Adding one moves the whole wave up by one.

In y = sin(2x), the angle changes twice as fast. One whole wave now fits into π radians. Read the start, height and repeat length from the full rule.

A minus sign in front turns a wave upside down: y = −sin(x). Its highs become lows. Check one known angle to see which way it goes.

Think you can tell the waves apart?

Choose sine or cosine graph questions. Check the start, height and repeat length. The game uses radians and may stretch or shift the basic wave.

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

Try sine graphs Try cosine graphs