cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two

English

If o.oangle x y = π/2, then cos(o.oangle x (x+y)) · ||x+y|| = ||x||.

Русский

Если o.oangle x y = π/2, то cos(угла между x и x+y) умноженное на норму x+y равно норме x.

LaTeX

$$$ o.oangle x y = \uparrow\left( \frac{\pi}{2} \right) \Rightarrow \cos\left(o.oangle x (x+y)\right) \cdot \|x+y\| = \|x\| $$$

Lean4

/-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
    Real.Angle.cos (o.oangle x (x + y)) * ‖x + y‖ = ‖x‖ :=
  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.cos_coe,
    InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]

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