Chapter 3.4, Problem 97E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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)

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