Derivative of cos(x)

Introduction

One of the fundamental trigonometric functions is the cosine function, denoted as cos(x), which represents the ratio of the length of the side adjacent 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 adjacent side OB in this scenario corresponds to the value of cos(x).

Small change in angle

Now, we will increase the angle by a small value dx as shown above, this will decrease the length of the adjacent side, the length of the new side will be cos(x+dx) represented by OD.

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 cos(x) is equivalent to the sine component of the arc length, Which is given by the line MA and it is equal to the line element DB.

Required Angle

To figure out Beginning 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 sin(x) component of the arc CA, constitutes the length of the change in the cos(x) side. Thus, -sin(x) dx signifies the decrease in the length of cos(x) side 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 equal to -sin(x).