InnerProductGeometry.cos_angle_add_of_inner_eq_zero
∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] {x y : V},
inner ℝ x y = 0 → Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x‖ / ‖x + y‖The cosine of an angle in a right-angled triangle as a ratio of sides.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Norm.normstatement and proof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- Inner.innerstatement and proof · cited by 1,089
- le_transproof · cited by 985
- norm_nonnegproof · cited by 725
- Real.cosstatement and proof · cited by 424
- InnerProductGeometry.anglestatement · cited by 170
- div_nonnegproof · cited by 103
- le_add_of_nonneg_rightproof · cited by 69
- mul_self_nonnegproof · cited by 35
Cited by5
Results whose statement or proof uses this declaration.
- InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zeroproof · cited by 3
- InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zeroproof · cited by 3
- EuclideanGeometry.cos_angle_of_angle_eq_pi_div_twoproof · cited by 2
- Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_twoproof · cited by 1
- InnerProductGeometry.cos_angle_sub_of_inner_eq_zeroproof · cited by 1