# Derivative of sine, geometric proof

Categories: differentiation calculus

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.

## Incrementing θ

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.

## Finding Δy

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:

## See also

- Slope of a curve
- Differentiation from first principles - x²
- Differentiation from first principles - a to the power x
- Second derivative and sketching curves
- Differentiation - the product rule
- Differentiation - the quotient rule
- Differentiation - the chain rule
- Differentiation - derivative of an inverse function
- Differentiation - L'Hôpital's rule

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