Trigonometry Function Formulas – Math Shortcut Tricks
Trigonometry Function Formulas
So, here are few Trigonometry Function Formulas. Let’s learn some basics of these formulas.
Trigonometry Function Formulas 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 | |
| -α | -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 sin 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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