2.4: Derivatives of the Trigonometric Functions

The motion of a Ferris wheel rotating at a constant speed provides an intuitive model for understanding trigonometric functions and their derivatives. As a rider moves along the circular path, the vertical height above the ground changes smoothly and periodically over time. This vertical motion can be accurately represented by a sine function, reflecting the repeating pattern of ascent and descent inherent to circular motion.

Height and Rate of Change

If the rider’s height is modeled by a sine function, the rate at which the height changes corresponds to the derivative of that function. This rate of change is not constant; instead, it varies smoothly and cyclically, reaching maximum values when the rider is moving most rapidly upward or downward and becoming zero at the highest and lowest points of the ride. Mathematically, the derivative of the sine function is the cosine function,

which captures this shifting rate of vertical motion.

Related Trigonometric Derivatives

The cosine function similarly describes periodic behavior, but with a phase shift relative to the sine function. Its derivative reflects how its values change over time and is given by

The negative sign indicates that the cosine function decreases where the sine function increases, and vice versa, highlighting the close relationship between these two functions.

The tangent function, defined as the ratio of sine to cosine, models quantities such as slope in periodic contexts. Its derivative follows directly from known trigonometric results and is expressed as

Together, these derivative relationships demonstrate how trigonometric functions describe both position and rate of change in systems involving smooth, periodic motion, such as a Ferris wheel rotating at a constant speed.

Consider a Ferris wheel rotating at a constant speed. The vertical height of a point on its edge changes smoothly over time, following a sine wave.

The rider's rate of change of height also varies smoothly and cyclically, like another wave. This changing rate matches the derivative of the sine function.

To find the derivative of sine x, start from the limit definition of a derivative and apply it to sine x.

First, expand the sine of (x + h) using the addition formula. Then, rearrange the terms and factor out sine x and cosine x.

As h approaches zero, cosine h minus 1 over h approaches zero, and sine h over h approaches 1.

Substituting these limits shows that the derivative of sine x is cosine x.

The same approach shows that the derivative of cosine x is negative sine x.

Since the tangent function is the ratio of sine to cosine, to find the derivative, apply the quotient rule. Now, substituting the values of the derivatives of the sine and cosine functions and simplifying using trigonometric identities gives the derivative of the tangent as secant squared.