3.5 HW Problems

Table of Contents:

Trigonometric Identities + Unit Circle (use as reference)

Derivative of the Sine Function

• To combine the derivative of $f(x)=\sin(x)$, for the x measured in radians we combine the limits with the angle sum identity for the sine function:

$$ \sin(x+h)=\sin(x)\cos(h)+\cos(x)\sin(h) $$

• So the derivative of the sine function is the cosine function:

$$ \frac{d}{dx}(\sin(x))=\cos(x) $$

Derivative of the Cosine Function

• With the help of the angle sum formula for the cosine function,

$$ \cos(x+h)=\cos(x)\cos(h)-\sin(x)\sin(h) $$

• So the derivative of the cosine function is the negative of the sine function:

$$ \frac{d}{dx}(\cos(x))=-\sin(x) $$