tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two

English

If o.oangle x y = π/2, then the angle between y and (y−x) equals arccos( ||y|| / ||y−x|| ).

Русский

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

LaTeX

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

Lean4

/-- The tangent of an angle in a right-angled triangle multiplied by the adjacent side equals
the opposite side. -/
theorem tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.tan (o.oangle x (x + y)) * ‖x‖ = ‖y‖ :=
  by
  have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
  rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
    InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]

Похожие утверждения