sin_oangle_sub_right_of_oangle_eq_pi_div_two

English

If o.oangle x y = π/2, then the sine of the angle between y and y−x equals ||x||/||y−x||.

Русский

Если o.oangle(x, y) = π/2, то синус угла между y и y−x равен ||x||/||y−x||.

LaTeX

$$$ \sin(o.oangle(y, y - x)) = \frac{\|x\|}{\|y - x\|} $$$

Lean4

/-- The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.sin (o.oangle y (y - x)) = ‖x‖ / ‖y - x‖ :=
  by
  have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
  rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
    InnerProductGeometry.sin_angle_sub_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))]

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