Sine and cosine, to be seen on the circle.
Sine and cosine look like two calculator keys with a table to learn by heart. In fact they are two lengths: the coordinates of a point turning on a circle of radius 1. Here you move the point yourself, and you watch sine, cosine, the right-angled triangle and the two waves change together, with the exact values when you stop on a special angle.
On a circle of radius 1, the cosine of an angle is the x coordinate of the point and the sine is the y coordinate: that is why sin²θ + cos²θ = 1, whatever θ is.
The principle
Sine and cosine are two lengths.
The interactive figure
Use the arrows on the figure (or swipe) to follow the reasoning one piece at a time, then drag the point P along the circle, or use the angle slider under the figure. On a keyboard: Tab reaches P, and there the arrow keys change the angle by one degree, Page Up / Page Down jump to the next special angle, Home and End take it to 0° and 360°. When focus is elsewhere in the figure, ← and → change step.
Hover (or tap) a part of the drawing — the point P, the sine, the cosine, the radius — to find out what it is.
A circle of radius 1 and a turning point
The circle has radius 1 and its centre at the origin. The point P starts on the right, on the x axis, and moves anticlockwise. The angle θ measures how far P has turned: drag P around the circle, or select it and use the arrow keys.
cos²θ + sin²θ = 0.59 + 0.41 = 1P is in the first quadrant
The reasoning, step by step
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Step 1 · The circle
A circle of radius 1 and a turning point
The circle has radius 1 and its centre at the origin. The point P starts on the right, on the x axis, and moves anticlockwise. The angle θ measures how far P has turned: drag P around the circle, or select it and use the arrow keys.
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Step 2 · The two shadows
Cosine and sine are the two shadows of P on the axes
From P drop straight down to the x axis: the distance from the origin is the cosine of θ. From P go across to the y axis: the distance from the x axis is the sine of θ. They are not formulas to remember, they are two lengths you read off the drawing, with their sign.
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Step 3 · The triangle
Radius, cosine and sine form a right-angled triangle
The cosine is horizontal, the sine vertical: they meet at a right angle. The radius closes them off and is the hypotenuse, which is always 1. So inside the circle there is a right-angled triangle with hypotenuse 1 and legs cos θ and sin θ.
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Step 4 · sin² + cos² = 1
Pythagoras on that triangle: why it is always 1
In a right-angled triangle the square of the hypotenuse is the sum of the squares of the legs. The hypotenuse is 1, so 1 = cos²θ + sin²θ. Move P wherever you like: the two legs change, the sum of their squares stays 1. It is Pythagoras’ theorem with the hypotenuse equal to 1.
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Step 5 · The signs
The signs tell you which quadrant you are in
The cosine is positive to the right of the y axis and negative to the left; the sine is positive above the x axis and negative below. In the four quadrants the pair (cosine, sine) has the signs (+, +), (−, +), (−, −), (+, −). On the axes one of the two is zero.
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Step 6 · The waves
The height of P, unrolled, draws a wave
As P turns, the graph on the right records its height as a function of the angle: out comes the sine wave. The cosine draws the same wave, but starts from 1 instead of 0. After a full turn, 360° that is 2π, everything starts again from the beginning.
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Step 7 · Try it
Special angles and two ways of measuring the angle
Drag P close to 30°, 45°, 60°, 90°: it snaps and the exact value appears. With the switch you go from degrees to radians: 180° is π, 90° is π/2. Turn the snapping off to look at all the angles in between, one by one with the arrow keys.
What each element of the drawing is
In the figure these explanations appear when you hover over the element. Here they are all written out.
- The point P
- It always sits on the circle of radius 1. Its coordinates are (cos θ, sin θ): reading the cosine and the sine of an angle means reading where P is.
- The cosine
- The x coordinate of P, that is its shadow on the horizontal axis. It is positive to the right of the origin, negative to the left, and never exceeds 1 in absolute value.
- The sine
- The y coordinate of P, that is its height above (or below) the horizontal axis. It is positive above the x axis, negative below, and never exceeds 1 in absolute value.
- The radius, hypotenuse of the triangle
- It is always 1: it is the hypotenuse of the right-angled triangle with legs cos θ and sin θ. By Pythagoras 1² = cos²θ + sin²θ, and that is why the sum of the squares never changes.
- The four quadrants
- Counted from the x axis, anticlockwise. In brackets the signs of (cosine, sine): first (+, +), second (−, +), third (−, −), fourth (+, −).
- The two waves
- In orange the sine, in blue the cosine, as functions of the angle. The dashed horizontal line carries the height of P straight to the tip of the sine wave.
Three questions to see if it has sunk in.
Try to answer by looking at the figure first, then open the answer.
1. P is in the third quadrant and its cosine is −0.6. What is the sine?
From the triangle, sin²θ = 1 − cos²θ = 1 − 0.36 = 0.64, so the sine is 0.8 in absolute value. In the third quadrant P is below the x axis, so the sine is negative: sin θ = −0.8.
2. Where is P at 5π/6, and what signs and exact values do sine and cosine have?
5π/6 is 150°: second quadrant, 30° from the negative x axis. The sine is positive and equals 1/2; the cosine is negative and equals −√3/2, that is the same numbers as at 30° with the sign of the quadrant.
3. Can there be an angle with sin θ = 1.2?
No. The sine is a leg of a right-angled triangle with hypotenuse 1, so it cannot exceed 1 (and cannot go below −1): P stays on the circle, and the maximum height of the circle is 1. If an exercise gives 1.2, there is a mistake upstream.
The questions we always get.
Why is the cosine the x coordinate and the sine the y, and not the other way round?
It is a convention, but with a reason: the angle is measured from the x axis, and in the right-angled triangle that P forms with that axis the cosine is the leg adjacent to the angle, the one resting on the x axis, while the sine is the opposite leg. With radius 1 the two legs become exactly the coordinates of P.
What does «one radian» mean on the circle?
An angle of one radian is the one that cuts off an arc of the circumference as long as the radius. Since the whole circumference measures 2π radii, the full turn is worth 2π radians: that is why 180° is π and 90° is π/2. Radians are just another way of measuring the same angle.
What happens beyond 360° or with a negative angle?
The point keeps turning and passes through the same places again: after a full turn sine and cosine take the same values again, which is why they are periodic functions with period 2π. A negative angle turns the other way, clockwise. The figure stops at one turn only to show the complete wave once.
Why do sine and cosine never exceed 1?
Because they are the legs of a right-angled triangle whose hypotenuse measures 1, and a leg can never be longer than the hypotenuse. You can see it in the drawing too: P never leaves the circle, so its height and its distance from the y axis stay between −1 and 1.
Keep going from here
Does trigonometry not add up for you?
When the circle becomes a drawing you can redo on your own, the formulas stop being a list to remember and become things you can work out. You catch up in a few hours, with the right exercises at the right time.