Derivative of sin(x)

Introduction

One of the fundamental trigonometric functions is the sine function, denoted as sin(x), which represents the ratio of the length of the side opposite to an angle x in a right-angled triangle to the length of the hypotenuse. We have to find the derivative of this function, Let us see how we can do this using a visual proof.

Consider a unit circle where the hypotenuse subtends an angle of x radians. The length of the opposite side AB in this scenario corresponds to the value of sin(x).

Small change in angle

Now, we will increase the angle by a small value dx as shown above, this will increase the length of the opposite side, the length of the new side will be sin(x+dx) represented by CD.

Arc length

To obtain the derivative we would need to isolate the change in the function, To do this we look at the arc length, which for very small angles approximates to a straight line and since it's length is equal to radius multiplied by angle subtended, the arc length will be equal to dx.

Thus, for a small change in the angle, the change in sin(x) is equivalent to the cosine component of the arc length, Which is given by the line CM.

Required Angle

To figure out the length of CM, We begin with the base angle x (AOB), it becomes apparent that the angle AOB and the alternate angle MAO are both equal to x, with angle CAM and angle MAO being complementary angles(since they are the angles between radius and tangent), angle CAM is equal to 90-x. Therefore, the required angle MCA is equal to x (Since the sum of all angles in a triangle is 180 degrees). Consequently, the cos(x) component of the arc CA, constitutes the length of the change in the sin(x) side. Thus, cos(x) dx signifies the small change in the sin(x) function when there is a small change in angle, dx.

Obtaining the Derivative Equation

Finally, the derivative is obtained by determining the ratio of the functional change to the angle change. This ratio, when calculated, yields the value of the derivative, which is equivalent to cos(x).