FM Pure Y2 Bicen

• What are the definitions of hyperbolic functions that need to be memorized?


  • \( \sinh x = \frac{e^x - e^{-x}}{2} \)
  • \( \cosh x = \frac{e^x + e^{-x}}{2} \)
  • \( \tanh x = \frac{\sinh x}{\cosh x} \)
• What shape does the graph of \( \sinh x \) resemble?


  It resembles a cubic shape
• What is the graph shape of \( \cosh x \)?


  It creates a U-shape crossing the y-axis at one
• What are the horizontal asymptotes of the graph of \( \tanh x \)?


  1 and -1
• What is the domain of \( \tanh x \)?


  It is only defined between -1 and 1
• What is the range of \( \cosh x \)?


  Always greater than or equal to one
• What are the definitions of the reciprocal hyperbolic functions?

  • \( \text{cosech} \, x = \frac{1}{\sinh x} \)
  • \( \text{sech} \, x = \frac{1}{\cosh x} \)
  • \( \text{coth} \, x = \frac{1}{\tanh x} \)
• What is the logarithmic equivalent of inverse hyperbolic functions?


  There is a logarithmic equivalent for these inverse hyperbolic functions
• What is the additional consideration when solving equations involving \( \text{arcosh} \)?


  There will be a plus and minus when solving equations
• What is Osborne's rule in relation to hyperbolic identities?


  • Negate any sine squareds from the trigonometric identity
  • For example, \( \cos^2 x - \sin^2 x \) becomes \( \cos^2 x - \sin^2 x \)
• How does the identity \( 1 + \tan^2 x = \sec^2 x \) change for hyperbolic functions?


  It becomes \( 1 - \tanh^2 x = \text{sech}^2 x \)
• What are the two methods for solving hyperbolic equations?


  1. Use identities and inverse functions
  2. Convert to exponential form and create a quadratic
• What is the Maclaurin expansion formula?


  It is \( f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \ldots \)
• How can standard results from the formula book be used in Maclaurin expansions?


  You can substitute into the standard results for compound functions
• What are the formulas for volumes of revolution in core pure 1?


  • \( V = \pi \int y^2 \, dx \) for rotation around the x-axis
  • \( V = \pi \int x^2 \, dy \) for rotation around the y-axis
• How does the parametric version of the volume formula differ from the standard version?


  It includes \( dt \) in the integrals
• What are the relationships between Cartesian and polar coordinates?


  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
  • \( r^2 = x^2 + y^2 \)
• What is the area formula in polar coordinates?


  It is \( \frac{1}{2} \int r^2 \, d\theta \) between \( \alpha \) and \( \beta \)
• How do areas in polar coordinates differ from non-polar coordinates?


  They resemble sectors like hands of a clock
• What are the rearranged double angle formulas for sine and cosine?


  • \( \sin^2 \theta = \frac{1}{2} - \frac{1}{2} \cos 2\theta \)
  • \( \cos^2 \theta = \frac{1}{2} + \frac{1}{2} \cos 2\theta \)
• What condition must be met for horizontal tangents in polar coordinates?


  It requires \( \frac{dy}{d\theta} = 0 \)