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Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: = | | | | = | | | | Next to the intersecting chords theorem and the tangent-secant theorem , the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle ...
There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, [4] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P. The angle θ between a chord and a tangent is half the arc belonging to the chord.
The word secant comes from the Latin word secare, meaning to cut. [2] In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points. [3]
The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.