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EnglandFurther Maths

Core Pure: polar coordinates and hyperbolic functions

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you find the area enclosed by a polar curve?

The area enclosed by a polar curve using the formula one half the integral of r squared with respect to theta, including areas between two curves and the area of one loop.
A focused answer to the OCR A-Level Further Mathematics A content on the area enclosed by a polar curve, covering the formula one half the integral of r squared with respect to theta, choosing the correct limits, finding the area of a single loop, and finding the area between two polar curves.
11 min answer →

How do you differentiate and integrate hyperbolic functions, and how do they help integrate certain algebraic functions?

Differentiation and integration of hyperbolic and inverse hyperbolic functions, and using hyperbolic substitutions to integrate functions involving the square root of x squared plus or minus a squared.
A focused answer to the OCR A-Level Further Mathematics A content on calculus with hyperbolic functions, covering the derivatives and integrals of sinh, cosh and tanh and their inverses, the standard integrals giving inverse hyperbolic functions, and using hyperbolic substitutions to integrate functions involving the square root of x squared plus or minus a constant.
12 min answer →

What are the hyperbolic functions, what identities do they satisfy, and what do their inverses look like?

The hyperbolic functions defined from exponentials, their graphs and properties, the key identities, and the logarithmic forms of the inverse hyperbolic functions.
A focused answer to the OCR A-Level Further Mathematics A content on hyperbolic functions, covering the definitions of sinh, cosh and tanh from exponentials, their graphs and odd or even properties, the key identities such as cosh squared minus sinh squared equals one, and the logarithmic forms of the inverse hyperbolic functions.
11 min answer →

How do polar coordinates describe points and curves, and how do you convert between polar and Cartesian form?

Polar coordinates, conversion between polar and Cartesian form, and sketching polar curves r = f(theta) including circles, lines, cardioids and spirals.
A focused answer to the OCR A-Level Further Mathematics A content on polar coordinates and curves, covering the polar representation of a point, conversion between polar and Cartesian coordinates, and sketching polar curves r = f(theta) such as circles, lines, cardioids and spirals, including finding where the curve meets the pole and its symmetry.
11 min answer →