InnerProductGeometry.sin_angle_add_mul_norm_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.sin (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖y‖The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the opposite side.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
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
- add_zeroproof · cited by 2,707
- MulZeroClass.mul_zeroproof · cited by 2,091
- Inner.innerstatement and proof · cited by 1,089
- Real.sinstatement and proof · cited by 389
- norm_zeroproof · cited by 366
- ne_of_ltproof · cited by 203
- InnerProductGeometry.anglestatement and proof · cited by 170
- div_mul_cancel₀proof · cited by 122
Cited by3
Results whose statement or proof uses this declaration.
- EuclideanGeometry.sin_angle_mul_dist_of_angle_eq_pi_div_twoproof · cited by 2
- Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_twoproof · cited by 1
- InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zeroproof · cited by 1