Beyond 90
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right-angled triangle the sine is simply the length of the opposite side as
a fraction of the hypotenuse. Angle O = 24ºHypotenuse, OA = 1Opposite, AB = 0·4 |
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And if the triangle is enlarged so that the hypotenuse is 2 units, but the angle still 24º, the opposite side will be 0·8, still 0.4 x hypotenuse.
Of course you cannot have a right-angled triangle with a second angle greater than, or equal to 90º. But it is possible to calculate the sine of angles greater than 90º. This becomes clear if the triangle is drawn on 2D coordinates. It also keeps the calculations simple if the ‘hypotenuse’ is kept 1 unit long.
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Doing this the sine is just the height of the triangle, the y-coordinate of point A. |
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So if OA is vertical, we no longer have a triangle, but we do have sin90 = 1 |
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| And if OA continues to rotate anti-clockwise around O until the angle is 156º we get sin156 = 0·4 |
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When the angle is 180º then sin180 = 0 |
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For angles greater than 180º the y-coordinate of A is negative. So sin204 = -0·4 |
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The sine of 336º is also -0·4 |
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Plotting the results for all angles between 0 and 360 gives the distinctive sine curve graph.
So although each angle has only one sine, the sine has two possible angles between 0º and 360º, but your calculator will only give you one of these. The second angle can be worked out with reference to a sketch of the graph and its symmetry.
| Eg. Which two angles between 0º and 360º have sin = 0·5? Doing INV sin 0·5 on a calculator will give 30º. Looking at the graph it can be seen that the curve between 0º and 180º is symmetrical about 90º and so the height of the graph at (180-30) is the same as its height at 30. So the second angle that has a sine of 0·5 is 150º And this can be checked by doing sin150 on a calculator. |
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