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Formulas: the plane and space, grade 11 (3 units)

Scale

  • The real length – l\displaystyle l the length on the map, at a scale of 1:n\displaystyle 1:n
    L\displaystyle L =\displaystyle = n⋅l\displaystyle n\cdot l
  • The real area – s\displaystyle s the area on the map, at a scale of 1:n\displaystyle 1:n
    S\displaystyle S =\displaystyle = n2⋅s\displaystyle n^{2}\cdot s

Similar shapes

  • The scale factor – a1\displaystyle a_1, a2\displaystyle a_2 corresponding sides
    k\displaystyle k =\displaystyle = a2a1\displaystyle \frac{a_2}{a_1}
  • The ratio of the areas of similar shapes
    S2S1\displaystyle \frac{S_2}{S_1} =\displaystyle = k2\displaystyle k^{2}
  • A height by its shadow – H\displaystyle H, S\displaystyle S the object and its shadow; h\displaystyle h, s\displaystyle s a stick and its shadow
    HS\displaystyle \frac{H}{S} =\displaystyle = hs\displaystyle \frac{h}{s}

Trigonometry in the right-angled triangle

  • Sine – a\displaystyle a the side opposite the angle, c\displaystyle c the hypotenuse
    sin⁡α\displaystyle \sin\alpha =\displaystyle = ac\displaystyle \frac{a}{c}
  • Cosine – b\displaystyle b the side adjacent to the angle
    cos⁡α\displaystyle \cos\alpha =\displaystyle = bc\displaystyle \frac{b}{c}
  • Tangent – the opposite side over the adjacent side
    tan⁡α\displaystyle \tan\alpha =\displaystyle = ab\displaystyle \frac{a}{b}
  • The angle from its tangent (and likewise sin⁡−1\displaystyle \sin^{-1}, cos⁡−1\displaystyle \cos^{-1})
    α\displaystyle \alpha =\displaystyle = tan⁡−1ab\displaystyle \tan^{-1}\frac{a}{b}
  • Pythagoras’ theorem
    a2+b2\displaystyle a^2+b^2 =\displaystyle = c2\displaystyle c^2
  • A gradient in percent – α\displaystyle \alpha the angle with the horizontal
    p\displaystyle p =\displaystyle = 100⋅tan⁡α\displaystyle 100\cdot\tan\alpha