Tangent Curve
The tangent is the length of the side opposite to an angle as a fraction of the length of the side adjacent to the angle. As the angle changes the lengths of both the opposite and adjacent sides change, if the hypotenuse remains 1 unit long. Because of this the shape of the tangent graph is different from the Sine and Cosine graphs.
| For angles of less than 45º, the opposite side is shorter than the adjacent side, so the tangent of the angle is less than 1. |
 tan 23 = 0·42 |
| When the angle is 45º, we have an isosceles triangle with the opposite and adjacent sides the same length, so tan 45 = 1 |
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For angles of more than 45º, the opposite side is longer than the adjacent side, so the tangent of the angle is more than 1. |
 tan47 = 1·07 tan89 = 57·29 |
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The tangent of 90º is undefined because you are trying to draw a triangle with two right angles. Also, the adjacent is 0 units long and you are trying to divide the opposite length by 0. |
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For angles between 90 and 180 the opposite side is positive while the adjacent side is negative and so the tangent is negative. |
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For angles between 180 and 270, both the opposite and adjacent sides are negative and so the tangent is positive |
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For angles between 270 and 360 the opposite side is negative, the adjacent side is positive and so the tangent is negative |
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So the graph of tan x is
The graph repeats every 180º, but like sine and cosine, each tangent has two angles between 0º and 360º
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Eg. tan x = 7 x = 81.9º (from calculator) x = 81.9 + 180 (from graph) x = 261.9º |
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