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# How can the angle ratios of a right triangle be approximated without a calculator

Thu 14 Jul, 2016 11:23 am
How does the deviation of subsequent angles for a right triagle affect ratios of a right triangle. For example, can the ratio of square root of 3:1:square root of 2 be a result of opposite angle sides tangent to the 90 degree angle being a deviation(or inverse fraction, if you will) of the angles relative to the ratio of 2:1:1 for a 45 degree triangle? My rather temorous assumption is that a 30 degree angle resulting in square root of 3 for its opposite side in a right triangle is the result of its deviation from 45, 15, being the factor results in square root of (1/3, the ratio of 15 to 45) being the opposite ratio in a 30, 60, 90 triangle.
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fresco

1
Fri 15 Jul, 2016 01:09 am
@metricphile,
Pythagoras applied to half an equilateral triangle (sides nominally 2 units) gives the ratio of sides 1:2:rt3.
That gives the values of the trig ratios for 30,60,90 degrees.
What else are you implying ?
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fresco

1
Fri 15 Jul, 2016 02:54 am
@metricphile,
NB Pythagoras applied to a 90;45;45 triangle, (equal sides 1 unit) gives the 1;1;rt2 ratios.

The trig mnemonic is the table
..0...............30............45........60..........90
(rt 0)/2......(rt 1)/2......(rt 2)/2.....(rt 3/2......(rt 4)/2____SINE
(rt 4)/2......(rt 3)/2......(rt 2)/2.....(rt 1)/2.....(rt 0)/2____COSINE
..zero..........(rt 3)/3..........1.............rt 3.........infinity____TANGENT
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