tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two

English

If o.oangle x y = π/2, then o.oangle (y−x) y equals arctan( ||x|| / ||y|| ).

Русский

Если o.oangle x y = π/2, то острий угол между (y−x) и y равен arctan( ||x|| / ||y|| ).

LaTeX

$$$ o.oangle x y = \uparrow\left( \frac{\pi}{2} \right) \Rightarrow o.oangle (y-x) y = \operatorname{Real.Angle.coe}\left( \operatorname{Real.arctan}\left( \frac{\|x\|}{\|y\|} \right) \right) $$$

Lean4

/-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ :=
  by
  rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
  rw [add_comm]
  exact (-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h

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