#### 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=f∘g 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)=cosu

Substitute u=π180θ in the above equation,

g′(h(θ))=cosπ180θ

Thus, the derivative g′(h(θ)) is g′(h(θ))=cosπ180θ.

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

h′(θ)=ddθ(π180θ)=π180

Thus, the derivative of h(θ) is h′(θ)=π180.

Substitute cosπ180θ for g′(h(θ)) and π180 for h′(θ) in equation (2),

ddθ(sinθ)=cosπ180θ(π180)=π180cosπ180θ=π180cosθ (Q θ°=π180θ rad)

Hence, ddθ(sinθ)=π180cosθ is proved.