   Chapter 3.4, Problem 97E

Chapter
Section
Textbook Problem

Use the Chain Rule to show that if θ is measured in degrees, then d d θ ( sin θ ) = π 180 cos θ (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: the differentiation formulas would not be as simple if we used degree measure.)

To determine

To show: If θ is measured in degrees, then ddθ(sinθ)=π180cosθ  by using chain rule.

Explanation

Result used: Chain Rule

If g is differentiable at x and f is differentiable at g(x), then the composite function F=fg defined by F(x)=f(g(x)) is differentiable at x and F is given by the product

F(x)=f(h(x))h(x) (1)

Proof:

The derivative ddθ(sinθ) is obtained as follows,

Since θ is measured in degree, θ°=π180θ rad.

ddθ(sinθ)=ddθ(sinπ180θ)

Let h(θ)=π180θ and g(u)=sinu where u=h(θ).

Apply the chain rule as shown in equation (1),

ddθ(sinθ)=g(h(θ))h(θ) (2)

The derivative g(h(θ)) is computed as follows,

g(h(θ))=g(u)=ddu(g(u))=ddu(sinu)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 