sinx + sin^2x +sin^3x =1
cos^6x-4cos^4x + 8 cos^2x = ?
sinx + sin^2x +sin^3x =1
cos^6x-4cos^4x + 8 cos^2x = ?
Last Activity: 14 Years ago
Draw a unit circle with center O. Let a central angle with initial side OP and terminal side OQ contain x radians (that is, the arc PQ has length x). Drop a perpendicular from Q to OP meeting it at R. Then OR = cos(x) and RQ = sin(x). If those directed line segments are up or to the right, the lengths are positive. If they are down or to the left, the lengths are negative.
Values at special angles:
x sin(x) cos(x) tan(x) cot(x) sec(x) csc(x)
0 0 1 0 --- 1 ---
/6 1/2 sqrt(3)/2 sqrt(3)/3 sqrt(3) 2 sqrt(3)/3 2
/4 sqrt(2)/2 sqrt(2)/2 1 1 sqrt(2) sqrt(2)
/3 sqrt(3)/2 1/2 sqrt(3) sqrt(3)/3 2 2 sqrt(3)/3
/2 1 0 --- 0 --- 1
2/3 sqrt(3)/2 -1/2 -sqrt(3) -sqrt(3)/3 -2 2 sqrt(3)/3
3/4 sqrt(2)/2 -sqrt(2)/2 -1 -1 -sqrt(2) sqrt(2)
5/6 1/2 -sqrt(3)/2 -sqrt(3)/3 -sqrt(3) -2 sqrt(3)/3 2
0 -1 0 --- -1 ---
More values at special angles:
x/10
/5
sin(x) (-1+sqrt[5])/4 sqrt(10-2 sqrt[5])/4
cos(x) sqrt(10+2 sqrt[5])/4 (1+sqrt[5])/4
tan(x) sqrt(1-2/sqrt[5]) sqrt(5-2 sqrt[5])
cot(x) sqrt(5+2 sqrt[5]) sqrt(1+2/sqrt[5])
sec(x) sqrt(2-2/sqrt[5]) -1+sqrt[5]
csc(x) 1+sqrt[5] sqrt(2+2/sqrt[5])
Use the above values and the identities below to obtain values of trigonometric functions of the following multiples of /10:
3/10 =
/2 -
/5,
2/5 =
/2 -
/10,
3/5 =
/2 +
/10,
7/10 =
/2 +
/5,
4/5 =
-
/5,
9/10 =
-
/10.
|sin(x)| <= 1,
|cos(x)| <= 1,
|sec(x)| >= 1,
|csc(x)| >= 1.
sec(x) = 1/cos(x),
csc(x) = 1/sin(x),
cot(x) = 1/tan(x),
tan(x) = sin(x)/cos(x),
cot(x) = cos(x)/sin(x).
sin(-x) = -sin(x),
cos(-x) = cos(x),
tan(-x) = -tan(x),
cot(-x) = -cot(x),
sec(-x) = sec(x),
csc(-x) = -csc(x).
sin(/2-x) = cos(x),
cos(/2-x) = sin(x),
tan(/2-x) = cot(x),
cot(/2-x) = tan(x),
sec(/2-x) = csc(x),
csc(/2-x) = sec(x).
sin(/2+x) = cos(x),
cos(/2+x) = -sin(x),
tan(/2+x) = -cot(x),
cot(/2+x) = -tan(x),
sec(/2+x) = -csc(x),
csc(/2+x) = sec(x).
sin(-x) = sin(x),
cos(-x) = -cos(x),
tan(-x) = -tan(x),
cot(-x) = -cot(x),
sec(-x) = -sec(x),
csc(-x) = csc(x).
sin(+x) = -sin(x),
cos(+x) = -cos(x),
tan(+x) = tan(x),
cot(+x) = cot(x),
sec(+x) = -sec(x),
csc(+x) = -csc(x).
sin(2+x) = sin(x),
cos(2+x) = cos(x),
tan(2+x) = tan(x),
cot(2+x) = cot(x),
sec(2+x) = sec(x),
csc(2+x) = csc(x).
sin2(x) + cos2(x) = 1,
tan2(x) + 1 = sec2(x),
1 + cot2(x) = csc2(x).
sin(x+y) = sin(x)cos(y) + cos(x)sin(y),
cos(x+y) = cos(x)cos(y) - sin(x)sin(y),
tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)],
cot(x+y) = [cot(x)cot(y)-1]/[cot(x)+cot(y)].
sin(x-y) = sin(x)cos(y) - cos(x)sin(y),
cos(x-y) = cos(x)cos(y) + sin(x)sin(y),
tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)],
cot(x-y) = [cot(x)cot(y)+1]/[cot(y)-cot(x)].
sin(2x) = 2 sin(x)cos(x),
cos(2x) = cos2(x) - sin2(x),
= 2 cos2(x) - 1,
= 1 - 2 sin2(x),
tan(2x) = [2 tan(x)]/[1-tan2(x)],
cot(2x) = [cot2(x)-1]/[2 cot(x)].
|sin(x/2)| = sqrt([1-cos(x)]/2),
|cos(x/2)| = sqrt([1+cos(x)]/2),
|tan(x/2)| = sqrt([1-cos(x)]/[1+cos(x)]),
tan(x/2) = [1-cos(x)]/sin(x),
= sin(x)/[1+cos(x)].
sin(3x) = 3 sin(x) - 4 sin3(x),
cos(3x) = 4 cos3(x) - 3 cos(x),
tan(3x) = [3 tan(x)-tan3(x)]/[1-3 tan2(x)].
sin(4x) = 4 sin(x)cos(x)[2 cos2(x)-1],
cos(4x) = 8 cos4(x) - 8 cos2(x) + 1.
sin(5x) = 5 sin(x) - 20 sin3(x) + 16 sin5(x),
cos(5x) = 16 cos5(x) - 20 cos3(x) + 5 cos(x).
sin(6x) = 2 sin(x)cos(x)[16 cos4(x) - 16 cos2(x) + 3],
cos(6x) = 32 cos6(x) - 48 cos4(x) + 18 cos2(x) - 1.
sin(nx) = 2 sin([n-1]x)cos(x) - sin([n-2]x),
cos(nx) = 2 cos([n-1]x)cos(x) - cos([n-2]x),
tan(nx) = (tan[(n-1)x]+tan[x])/(1-tan[(n-1)x]tan[x]).
sin(x)cos(y) = [sin(x+y) + sin(x-y)]/2,
cos(x)sin(y) = [sin(x+y) - sin(x-y)]/2,
cos(x)cos(y) = [cos(x-y) + cos(x+y)]/2,
sin(x)sin(y) = [cos(x-y) - cos(x+y)]/2.
sin(x) + sin(y) = 2 sin[(x+y)/2]cos[(x-y)/2],
sin(x) - sin(y) = 2 cos[(x+y)/2]sin[(x-y)/2],
cos(x) + cos(y) = 2 cos[(x+y)/2]cos[(x-y)/2],
cos(x) - cos(y) = -2 sin[(x+y)/2]sin[(x-y)/2],
tan(x) + tan(y) = sin(x+y)/[cos(x)cos(y)],
tan(x) - tan(y) = sin(x-y)/[cos(x)cos(y)],
cot(x) + cot(y) = sin(x+y)/[sin(x)sin(y)],
cot(x) - cot(y) = -sin(x-y)/[sin(x)sin(y)].
[sin(x)+sin(y)]/[cos(x)+cos(y)] = tan[(x+y)/2],
[sin(x)-sin(y)]/[cos(x)+cos(y)] = tan[(x-y)/2],
[sin(x)+sin(y)]/[cos(x)-cos(y)] = -cot[(x-y)/2],
[sin(x)-sin(y)]/[cos(x)-cos(y)] = -cot[(x+y)/2],
[sin(x)+sin(y)]/[sin(x)-sin(y)] = tan[(x+y)/2]/tan[(x-y)/2].
sin2(x) - sin2(y) = sin(x+y)sin(x-y),
cos2(x) - cos2(y) = -sin(x+y)sin(x-y),
cos2(x) - sin2(y) = cos(x+y)cos(x-y).
sin2(x) = (1 - cos[2x])/2,
cos2(x) = (1 + cos[2x])/2,
tan2(x) = (1 - cos[2x])/(1 + cos[2x]),
sin3(x) = (3 sin[x] - sin[3x])/4,
cos3(x) = (3 cos[x] + cos[3x])/4,
sin4(x) = (3 - 4 cos[2x] + cos[4x])/8,
cos4(x) = (3 + 4 cos[2x] + cos[4x])/8,
sin5(x) = (10 sin[x] - 5 sin[3x] + sin[5x])/16,
cos5(x) = (10 cos[x] + 5 cos[3x] + cos[5x])/16,
sin6(x) = (10 - 15 cos[2x] + 6 cos[4x] - cos[6x])/32,
cos6(x) = (10 + 15 cos[2x] + 6 cos[4x] + cos[6x])/32,
Last Activity: 14 Years ago
dear friend,
kindly send your address, so that i'll send the solutions by post
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