Trigonometry Function Formulas

Trigonometry Function Formulas

Trigonometric Function of a Right Triangle :

sin α = a / c = opposite / hypotenuse

cos α =  b / c = adjacent / hypotenuse

tan α = a / b = opposite / adjacent

cot α = b / a = adjacent / opposite

 

sec α = c / b                       Cosec α = c / a

 

Basic Formula :

sin2 α + cos2 α = 1

tan α . cot tan α = 1

tan α = sin α / cos α = 1 / cot tan α

cot tan α = cos α / sin α = 1 / tan α

1 + tan2 α = 1 / cos2 α = sec2 α

1 + cot tan2 α = 1 /  sin2 α = cos sec2 α

 

Trigonometric Table

α 00 300 450 600 900 1200 1800 2700 3600
sin α 0 1/2 √2/2 √3/2 1 √3/2 0 -1 0
cos α 1 √3/2 √2/2 1/2 0 -1/2 -1 0 1
tan α 0 1/√3 1 √3 -√3 0 0
cot α √3 1 1/√3 0 -1/√3 0
sec α 1 2/√3 √2 2 -2 -1 1
cosec α 2 √2 2/√3 1 2/√3 -1

Co-Ratios

sin cos tan cot
-sin α +cos α -tan α -cot α
900 – α +cos α +sin α +cot α +tan α
900 + α +cos α -sin α -cot α -tan α
1800 – α +sin α -cos α -tan α -cot α
1800 + α -sin α -cos α +tan α +cot α
2700 – α -cos α -sin α +cot α +tan α
2700 + α -cos α +sin α -cot α -tan α
3600k – α -sin α +cos α -tan α -cot α
3600k – α +sin α +cos α +tan α +cot α

 

 

Trigonometry Addition Formula:

  • sin(A + B) = sinA cosB + cosA sinB
  • sin(A – B) = sinA cosB – cosA sinB

 

  • cos(A + B) = cosA cosB – sinA sinB
  • cos(A – B) = cosA cosB + sinA sinB

 

  • tan (A + B) = tanA + tanB / 1 – tanA tanB
  • tan(A – B) =  tanA – tanB / 1 + tanA tanB

 

  • cot (A+ B) = cotA cotB – 1 / cotA + cotB

 

 

Product of Trigonometric Functions:

 

  • sin α cos β = 1/2 [ sin (α + β) + sin(α – β)]
  • cos α cos β = 1/2 [ sin (α + β) + sin(α – β)]
  • cos α cos β = 1/2 [ cos (α + β) + cos(α – β)]
  • sin α sin β = 1/2 [ cos (α – β) + cos(α + β)]

 

  • tan α tan β = tan α + tan β / cot tan α + cot tanβ = – tanα – tan β / cot tan α – cot tan β

 

 

Trigonometric Formula with t = tan(x/2)

sinx = 2t / 1 + t2

cos x = 1 – t2 / 1 + t2

tan x = 2t / 1 – t2

cot x = 1 – t2 / 2t

 

 

Trigonometric Relation Between Functions:

 

 

Angle of a Plane Triangle :

  • A, B, C are 3 angles of a triangle
  • sin A + sin B + sin c = 4 cos(A / 2) cos(B/2) cos(C/2)
  • cosA + cos B + cos C = 4 sin(A/2) sin(B/2) sin(C/2) + 1
  • sinA + sinB – sinC = 4sin (A/2) sin (B/2) cos (C/2)

 

 

 

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