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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