Trigonometric Derivatives Discussion

This discussion builds comfort with derivatives of the trigonometric functions by asking you to think about some applications and consequences of them. Please post at least one response to at least one of the following by class time Monday (October 5).

Applications

Lots of real-world phenomena that involve circles or triangles have mathematical descriptions that involve trigonometric functions. Derivatives of those functions often show up in problems involving speeds or accelerations (since speed is the derivative of position and acceleration is the derivative of velocity). For example...

A rider on a carnival ferris wheel with radius 12 feet will have a height above ground level (or at least boarding level), in feet, of

where t represents time, measured in units of however long it takes the ferris wheel to rotate 1 radian. What is the rider’s vertical speed as a function of time? Vertical acceleration?

There is an old farm silo behind my house, which I estimate to be about 30 feet high. When the sun is rising, this silo casts a shadow across the ground whose length in feet, L, is given by the equation

where Θ is the sun’s angle above the ground in radians. How “fast” is the length of this shadow changing, as a function of Θ? (Later in this course we may come back to this silo, when we know how to calculate how fast the shadow is changing in more meaningful units involving actual time.)

Antiderivatives

Now that we have differentiation rules for trigonometric functions, we automatically have antiderivative rules also, at least where it is reasonable to “run” the differentiation rules backwards. For example...

What is the antiderivative of cos x?

What is the antiderivative of -sin x? How about sin x?

What other antiderivatives can you evaluate with other trigonometric differentiation rules?

Higher-Order Derivatives

(These questions are inspired by material in “Higher-Order Derivatives” in section 3.5 of our textbook. But don’t peek! That takes all the fun out of making discoveries for yourself.)

What are the second, third, and fourth derivatives of sin x?

What are the second, third, and fourth derivatives of cos x?

Is there a pattern to the higher-order derivatives of sine and cosine? Can you describe that pattern in words (or equations)?

What is the 123^rd derivative of sin x?