Category

Trigonometric differentiation

Summary
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation. The diagram at right shows a circle with centre O and radius r = 1. Let two radii OA and OB make an arc of θ radians. Since we are considering the limit as θ tends to zero, we may assume θ is a small positive number, say 0 < θ < 1⁄2 π in the first quadrant. In the diagram, let R1 be the triangle OAB, R2 the circular sector OAB, and R3 the triangle OAC. The area of triangle OAB is: The area of the circular sector OAB is , while the area of the triangle OAC is given by Since each region is contained in the next, one has: Moreover, since sin θ > 0 in the first quadrant, we may divide through by 1⁄2 sin θ, giving: In the last step we took the reciprocals of the three positive terms, reversing the inequities. We conclude that for 0 < θ < 1⁄2 π, the quantity sin(θ)/θ is always less than 1 and always greater than cos(θ). Thus, as θ gets closer to 0, sin(θ)/θ is "squeezed" between a ceiling at height 1 and a floor at height cos θ, which rises towards 1; hence sin(θ)/θ must tend to 1 as θ tends to 0 from the positive side:For the case where θ is a small negative number –1⁄2 π < θ < 0, we use the fact that sine is an odd function: The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of θ is unimportant.