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Precalculus Unit 3 Formulas and Concepts

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  • Law of Sines formula
    The Law of Sines states \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\) relating sides and angles of a triangle.
  • Law of Cosines formula for side a
    \(a^2 = b^2 + c^2 - 2bc \cdot \cos A\) calculates side a using sides b, c and angle A.
  • Heron's Formula for triangle area
    Area \(K = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \frac{a+b+c}{2}\) is the semi-perimeter.
  • Polar coordinate conversion for r
    \(r = \sqrt{x^2 + y^2}\) gives the radius from Cartesian coordinates.
  • Polar coordinate conversion for x
    \(x = r \cos \theta\) converts polar to Cartesian x-coordinate.
  • Polar coordinate conversion for y
    \(y = r \sin \theta\) converts polar to Cartesian y-coordinate.
  • Tangent in polar coordinates
    \(\tan \theta = \frac{y}{x}\) relates angle to Cartesian coordinates.
  • Vector magnitude formula
    The magnitude of vector v with components a and b is \(||\mathbf{v}|| = \sqrt{a^2 + b^2}\).
  • Vector from direction and magnitude
    Vector v can be found as \(\mathbf{v} = v \cos \alpha \mathbf{i} + v \sin \alpha \mathbf{j}\).
  • Equation of a parabola (vertical axis)
    \(y^2 = 4ax\) represents a parabola opening right or left.
  • Equation of a parabola (horizontal axis)
    \(x^2 = 4ay\) represents a parabola opening up or down.
  • Standard form of ellipse
    \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) describes an ellipse centered at the origin.
  • Standard form of ellipse (vertical major axis)
    \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\) ellipse with vertical major axis.
  • Relationship between ellipse axes and focal distance
    \(a^2 - b^2 = c^2\) relates semi-major axis a, semi-minor axis b, and focal distance c.
  • Standard form of hyperbola (horizontal transverse axis)
    \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) describes a hyperbola centered at the origin.
  • Standard form of hyperbola (vertical transverse axis)
    \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) hyperbola with vertical transverse axis.
  • Relationship between hyperbola axes and focal distance
    \(a^2 + b^2 = c^2\) relates transverse axis a, conjugate axis b, and focal distance c.
  • Coordinates of key angles on the unit circle
    Key angles like 0°, 90°, 180°, 270° correspond to points (1,0), (0,1), (-1,0), (0,-1) on the unit circle.
  • Coordinates of 45° on the unit circle
    At 45° (or \(\frac{\pi}{4}\)), coordinates are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).
  • Coordinates of 30° and 60° on the unit circle
    At 30° (\(\frac{\pi}{6}\)) coordinates are \((\frac{\sqrt{3}}{2}, \frac{1}{2})\), and at 60° (\(\frac{\pi}{3}\)) coordinates are \((\frac{1}{2}, \frac{\sqrt{3}}{2})\).