Home > Justincase > A geometric proof of trigonometric derivatives

## A geometric proof of trigonometric derivatives

As we already know, the first derivatives of $\sin(x)$ and $\cos(x)$ are $\cos(x)$ and $-\sin(x)$ respectively. The question is how do we prove it?

Since high school, the only proof that I knew is to use two of the trigonometric identities which are:

• $\sin \alpha - \sin \beta = 2 \sin (\frac{\alpha-\beta}{2}) \cos (\frac{\alpha+\beta}{2})$, and
• $\cos \alpha - \cos \beta = -2 \sin (\frac{\alpha-\beta}{2}) \sin (\frac{\alpha+\beta}{2})$.

Hence,

$\frac{d}{dx}\sin x = \displaystyle{\lim_{\Delta x \rightarrow 0} \frac{\sin(x+\Delta x)-\sin(x)}{\Delta x} = \lim_{\Delta x \rightarrow 0} 2 \sin(\frac{\Delta x}{2})\cos(\frac{2x+\Delta x}{2})=\cos x}$.

The derivative of $\cos x$ can also be derived using the second identity.

However, there is another proof, which is (I think) more elegant. It is using a geometric interpretation of $\sin x$ and $\cos x$. Let us draw a unit circle in $\mathbb{R}^2$.

Thus we have $y=\sin \theta$ and $x=\cos \theta$. We are looking for $\frac{dy}{d \theta}$ and$\frac{dx}{d \theta}$, however we are not going to discuss the first derivative of $\cos \theta$ as it can be done in the same way as $\sin \theta$.

$\displaystyle{\frac{d \sin \theta}{d \theta}=\lim_{\Delta \theta \rightarrow 0} \frac{\sin(\theta + \Delta \theta)-\sin(\theta)}{\Delta \theta}=\lim_{\Delta \theta \rightarrow 0} \frac{y+\Delta y - y}{\Delta \theta}=\lim_{\Delta \theta \rightarrow 0} \frac{\Delta y}{\Delta \theta}}.$

We are now going to estimate $\Delta y$, see the second figure.  The following relationship applies:

$\sin \phi = \frac{\Delta y}{r}$. or $\Delta y = r \sin \phi$.

Thus,

$\displaystyle{\frac{d \sin \theta}{d \theta}=\lim_{\Delta \theta \rightarrow 0} \frac{\Delta y}{\Delta \theta}=\lim_{\Delta \theta \rightarrow 0} \sin \phi \frac{r}{\Delta \theta}}$,

where $r$ is the chord PQ and $\Delta \theta$ is the arc PQ. As $\Delta \theta$ goes to zero, the ratio of the arc PQ and the chord PQ tends to 1 and the chord PQ tends to become a tangent line of the unit circle at $(x,y)$, therefore $\sin \phi = \sin (\theta + \pi/2)$. Assuming that the limit of multiplication is the same as the multiplication of limit, we have the following:

$\frac{d \sin \theta}{d \theta}=\sin(\theta + \pi/2) = \cos(\theta)$.

As mentioned before, the first derivative of $\cos \theta$ can be done the same way.

Also, I should thank Pak Yudi Soeharyadi as I got the idea of this geometric proof from him.

Categories: Justincase
1. March 14, 2013 at 12:34 pm

Nice to get to know your all posts and surfing inside!

• March 15, 2013 at 1:07 am

Thank you.. It is also nice to read yours. Quite motivational.

-salam hangat dari Tangerang-

2. May 3, 2013 at 3:28 am

I do accept as true with all the ideas you’ve presented in your post. They are very convincing and will certainly work. Nonetheless, the posts are too quick for newbies. May just you please extend them a bit from subsequent time? Thanks for the post.

3. May 2, 2017 at 1:15 am

I think your second-listed identity should read $\cos\alpha-\cos\beta = -2 \sin\left(\frac{\alpha-\beta}{2}\right) \sin\left(\frac{\alpha+\beta}{2}\right)$

• May 2, 2017 at 1:34 am

Thanks for the comment. I edited it.

4. March 11, 2018 at 4:05 pm

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5. March 11, 2018 at 4:11 pm

also sure here we must consider some aspects and considerations as say prpf dr mircea orasanu and prof horia orasanu concerning as followed
FIRST DERIVATIVE AND FERMAT , LAGRANGE AND DARBOUX THEOREMS
ABSTRACT
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object’s velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

6. March 16, 2018 at 4:28 am

here we consider the some and sure we see as say prof dr mircea orasanu and prof horia orasanu as followed
DERIVATIVE OF FUNCTION AND GEOMETRIC INTERPRETATION
FERMAT THEOREM

ABSTRACT
et AB be the secant line, passing through the points and . If , that is, approaches zero, then the secant line approaches the tangent line at the point . Accordingly, the slope of the tangent line is the limit of the slope of the secant line when approach zero:

Thus, the derivative can be interpreted as the slope of the tangent line at the point on the graph of the function .

If a function is increasing on some interval, then the slope of the tangent is positive at each point of that interval, and hence, the derivative of the function is positive.

If a function is decreasing on some interval, then the slope of the tangent is negative at each point of that interval, and hence, the derivative of the function is negative.

If a function is increasing on some interval, then the slope of the tangent is positive at each point of that interval, and hence, the derivative of the function is positive.

If a curve y = f (x) has a smooth top then the peak of the curve serves as the boundary between intervals of increasing and decreasing of the function. At this point the tangent is parallel to the x-axis. Therefore, its slope equals zero, and so .

Likewise, if a curve y = f (x) has a smooth bottom then there exists a point peak of the curve serves as the boundary between intervals of increasing and decreasing of the function. At this point the tangent is parallel to the x-axis. Therefore, its slope equals zero, and so .

7. March 18, 2018 at 3:36 pm

here we mention an important fact as say prof dr mircea orasanu and prof horia orasanu concerning
GEOMETRY OF SECOND DERIVATIVES OF FUNCTIONS
AND DIFFERENTIAL APPLICATION
ABSTRACT
The second derivative of a function is the derivative of the first derivative and it indicates the change in gradient of the original function. The sign of the second derivative tells us if the gradient of the original function is increasing, decreasing or remaining constant.

To determine the second derivative of the function f(x)
, we differentiate f′(x) using the rules for differentiation.
1 INTRODUCTION
Often the most confusing thing for a student introduced to differentiation is the notation associated with it. Here an attempt will be made to introduce as many types of notation as possible.

A derivative is always the derivative of a function with respect to a variable. When we write the definition of the derivative as we mean the derivative of the function f(x) with respect to the variable x.

One type of notation for derivatives is sometimes called prime notation. The function f´(x), which would be read “f-prime of x”, means the derivative of f(x) with respect to x. If we say y = f(x), then y´ (read “y-prime”) = f´(x). This is even sometimes taken as far as to write things such as, for y = x4 + 3x (for example), y´ = (x4 + 3x)´. This notation suggests that perhaps derivatives can be treated like fractions, which is true in limited ways in some circumstances.