I have a table of natural sines, cosines & tangent. There are columns and rows upto degree 90 and mean difference etc. I can find the value of sine of any angle with the help of this table if i dont have any calculator. My Questions are:
1. if sine 30 degree = 0.5 (from the table), how can I calculate reverse, i.e., how can i calculate for 0.5 so that I get 30 degree. I haven't got any cosec table for this. Is there any table for inverse funtions.
2. where do these values in tables come from ? What are the procedure to find out such values for any angle upto 90 degree.
3. In calculator, I press 30, then sin and get 0.5, and then press inverse and sin^-1 and get 30. How does this last calcuation is performed in calculator.?
1. if sine 30 degree = 0.5 (from the table), how can I calculate reverse, i.e., how can i calculate for 0.5 so that I get 30 degree. I haven't got any cosec table for this. Is there any table for inverse funtions.
2. where do these values in tables come from ? What are the procedure to find out such values for any angle upto 90 degree.
3. In calculator, I press 30, then sin and get 0.5, and then press inverse and sin^-1 and get 30. How does this last calcuation is performed in calculator.?
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1) If sin(30) = 0.5, then arcsin(0.5) = 30. If you want to go backwards, simply go the other way on the table you have. You're thinking too much into it.
2) There are multiple trig identities that help find exact values for angles. You can take a half of sin(30).
sin(15) = sin(45 - 30)
sin(15) = sin(45)cos(30) - cos(45)sin(30)
sin(15) = √(2)/2*√(3)/2 - √(2)/2*1/2
sin(15) = (√6 - √2) / 4
Archimedes was that man that supposedly completed the angles table. What he did was what I just did up there. When he got close to an angle of 1, he approximated sin(1) with a line. He then reversed what I did to find sin(2), sin(3), so on. If you want to know how to do this, keep taking math. They'll probably teach you it in grade 12.
3) Your calculator does trigonometry by using a Taylor approximation. If you want to know how THIS is done, you'll have to take Calculus in university.
2) There are multiple trig identities that help find exact values for angles. You can take a half of sin(30).
sin(15) = sin(45 - 30)
sin(15) = sin(45)cos(30) - cos(45)sin(30)
sin(15) = √(2)/2*√(3)/2 - √(2)/2*1/2
sin(15) = (√6 - √2) / 4
Archimedes was that man that supposedly completed the angles table. What he did was what I just did up there. When he got close to an angle of 1, he approximated sin(1) with a line. He then reversed what I did to find sin(2), sin(3), so on. If you want to know how to do this, keep taking math. They'll probably teach you it in grade 12.
3) Your calculator does trigonometry by using a Taylor approximation. If you want to know how THIS is done, you'll have to take Calculus in university.
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I'm not sure what you mean by reverse. You mention cosecant, but
1 / sin(x) = csc(x)
for all x (≠ πn or 180ºn if you prefer degrees). That is a well-known identity. If you mean that you know that sin(α) = 1/2 for some angle α and you want to find α, then just look at the table backwards. Go down the sin(α) column and find which gives 1/2, or 0.5, and then look at which angle gave that answer.
As for how your calculator does this stuff, some have these tables in memory, some use Taylor series (polynomial approximations of the sine curve).
"Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy."
-- http://en.wikipedia.org/wiki/Trigonometr…
1 / sin(x) = csc(x)
for all x (≠ πn or 180ºn if you prefer degrees). That is a well-known identity. If you mean that you know that sin(α) = 1/2 for some angle α and you want to find α, then just look at the table backwards. Go down the sin(α) column and find which gives 1/2, or 0.5, and then look at which angle gave that answer.
As for how your calculator does this stuff, some have these tables in memory, some use Taylor series (polynomial approximations of the sine curve).
"Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy."
-- http://en.wikipedia.org/wiki/Trigonometr…