Derivatives of Sine and Cosine through Infinitesimals

The video explores the derivatives of sine and cosine functions, initially attempting to use first principles but then shifting to a geometric approach. The geometric proof involves the unit circle and similar triangles to derive the derivatives of sine and cosine. The video also discusses linearity properties, the product rule, and introduces the chain rule for composite functions.

What is the initial approach mentioned for finding the derivative of sine x?

• Using substitution

• Using the chain rule

• Using first principles

• Using integration

In the geometric proof, what is the significance of the unit circle?

• It is used to calculate the area of a circle.

• It helps in solving quadratic equations.

• It helps in defining the sine and cosine functions.

• It is used to find the length of a tangent.

What does the radius of a circle touching a tangent line represent?

• It bisects the tangent line.

• It is parallel to the tangent line.

• It is perpendicular to the tangent line.

• It is equal to the tangent line.

What is the formula for the arc length of a circle in terms of radius r and central angle alpha?

• r + alpha

• r / alpha

• r * alpha

• r - alpha

What does the derivative of sine theta represent in the geometric proof?

• The length of the hypotenuse

• The rate of change of sine with respect to theta

• The area of the triangle

• The circumference of the circle

How are similar triangles used in the geometric derivation of the derivative of sine?

• To calculate the circumference

• To determine the ratio of sides involving infinitesimals

• To find the area of the circle

• To solve quadratic equations

What is the derivative of sine theta?

• cosine theta

• sine theta

• negative cosine theta

• negative sine theta

What adjustment is made to find the derivative of cosine theta?

• Multiplying by a constant

• Switching the sign to negative

• Dividing by a constant

• Adding a constant

What is the derivative of cosine theta?

• negative cosine theta

• cosine theta

• negative sine theta

• sine theta

What is the chain rule used for in calculus?

• To find the derivative of a sum

• To find the derivative of a product

• To find the derivative of a composite function

• To find the integral of a function