# Basic Trigonometry Formulas & Identities

Topic Covered

### Six Important Trigonometric Ratios

There are six important trigonometric ratios (trigonometric functions) used in trigonometry: sine, cosine, secant, cosecant, tangent, and cotangent. In short, these are written as sin, cos, sec, csc, tan, and cot.

sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
cot θ = Adjacent / Opposite
cosec θ = Hypotenuse / Opposite
sec θ = Hypotenuse / Adjacent

### Reciprocal Trigonometry Formulas

sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
sec θ = 1/cos θ
cot θ = 1/tan θ
cosec θ = 1/sin θ

### Pythagorean Identities

sin²θ + cos²θ = 1
tan2θ + 1 = sec2θ
cot2θ + 1 = cosec2θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ

### Sum and Difference identities

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
tan(a + b) = (tan a + tan b) / (1 – tan a • tan b)
sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
tan (a − b) = (tan a – tan b) / (1 + tan a • tan b)

### Even and Odd Formulas

sin (-a) = – Sin a
cos (-a) = Cos a
tan (-a) = -Tan a
csc (-a) = – Csc a
sec (-a) = Sec a
cot (-a) = -Cot a

### Double-Angle Formula

sin(2x) = 2sin(x)cos(x) = 2tanx / (1+tan²x)
cos(2x) = cos²(x) – sin²(x) = (1−tan²x) / (1+tan²x)
cos(2x) = 2cos²(x) −1 = 1 – sin²(x)
tan(2x) = 2tan(x) / 1−tan²(x)
sec(2x) = sec²x(2−sec²x)
csc(2x)=(secx.cscx) / 2

### Inverse Trigonometry Formulas

sin-1 (–x) = – sin-1 x
cos-1 (–x) = π – cos-1 x
tan-1 (–x) = – tan-1 x
cosec-1 (–x) = – cosec-1 x
sec-1 (–x) = π – sec-1 x
cot-1 (–x) = π – cot-1 x

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