The first derivative of the sine function is the cosine function:
In this article, we will provide a simple geometric proof of this result.
Sine in a unit circle
This diagram shows part of a unit circle, centred at O:
The right-angled triangle OPA has an angle θ at the centre of the circle. We can find the sine of the angle from the standard formula:
We will represent the length AP as y. Since OA is the radius of a unit circle, it is 1. So the sine of θ is simply:
While we are looking at this diagram, it is also worth noting that the arc of the circle between X and A has a length θ. That is because the arc subtends an angle θ at the centre of the circle. The length of an arc is given by rθ (provided θ is measured in radians), and since r is 1 for a unit circle, the length is simply θ. We will use this result later.
Here is what happens when we increase θ by a small amount Δθ:
If we draw a radius with an angle of θ + Δθ, it meets the circumference as point B.
We previously noted that AP, the perpendicular height of A from the x-axis, is equal to the sine of θ. We called the distance y:
The perpendicular height of B from the x-axis is BQ. Let's call this new distance y + Δy. This is equal to the sine of θ + Δθ:
We can now write a difference formula for the change in sine θ, as θ changes:
The limit as Δθ tends to zero gives us the derivative of the sine function:
If we can find an expression for Δy as a function of θ, we should be able to solve this equation to find the derivative of the sine function.
We will now add an extra point C to create a triangle ABC. We will place C vertically above point A and horizontally in line with point B:
Some observations about the triangle ABC:
- Since AC is vertical and BC is horizontal, angle C must be a right angle.
- Since A is a distance y from the x-axis, and B is a distance y + Δy from the x-axis, side AC has length Δy.
- The arc length AB has length Δθ for reasons we discussed earlier.
Triangle ABC as Δθ tends to zero
As Δθ tends to zero, the arc AB tends towards a straight line, and furthermore, that straight line is a tangent to the circle:
In the limit as AB tends towards a straight line, the length of the line AB tends towards the length of the arc AB, which is Δθ.
Also, since a tangent and a radius of a circle meet at 90 degrees, the angle OAB tends towards a right angle.
It is easy to verify through simple geometry that, when OAB is a right angle, the angle BAC is equal to the angle AOP, which is θ.
Finding the derivative of sine
Here is the triangle ABC, and everything we know about it:
This represents the triangle as Δθ tends to zero. From the triangle, basic trigonometry tells us that:
We know from earlier that:
This proves that:
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