tangent

In graphs, a straight line that touches a curve and gives the gradient of the curve at the point of contact. At a maximum, minimum, or point of inflection, the tangent to a curve has zero gradient.

To find the gradient of the tangent of the curve at point P, extend the tangent AC and construct the right-angled triangle ABC:

The gradient of the tangent at the point P is AB/BC. This is only an approximation. To make the answer as accurate as possible, the length of the tangent AC should be large.

For example, to find the gradient of the curve y = x² − 2x at the point P(2,0):

P = AB/BC = 5/2.5 = 2

See also (trigonometry) and (circle).

tangent

The tangent of an angle is a mathematical function used in the study of right-angled triangles. If the tangent of an angle β is known, then the length of the opposite side can be found given the length of the adjacent side, or vice versa.

In trigonometry, a function of an acute angle in a right-angled triangle, defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to it; a way of expressing the gradient of a line. This function can be used to find either sides or angles in a right-angled triangle.

For example, to work out the height (h) of a 5-m ramp set at an angle of 8°:

tan 8° = opposite/adjacent = h/5

h = 5 × tan 8° = 0.70 m (to the nearest cm)

The height of the ramp is 70 cm.

See also sine, cosine, tangent (graph), and tangent (circle).

tangent

Line that touches a circle at only one point. A tangent is at right angles to the radius at the point of contact. From any point outside a circle, the lines of two tangents drawn to the circle will be of equal length.

See also circle theorems, tangent (trigonometry), and tangent (graph).