Download TRIGNOMETRY FORMULAS & IDENTITIES and more Cheat Sheet Mathematics in PDF only on Docsity! TRIGNOMETRY FORMULAS & IDENTITIES:- Basic Trigonometric Function Formulas: There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent. By using a right-angled triangle as a reference, the trigonometric functions and identities are derived: • sin θ = Opposite Side/Hypotenuse • cos θ = Adjacent Side/Hypotenuse • tan θ = Opposite Side/Adjacent Side • sec θ = Hypotenuse/Adjacent Side • cosec θ = Hypotenuse/Opposite Side • cot θ = Adjacent Side/Opposite Side Reciprocal Identities: The Reciprocal Identities are given as: • cosec θ = 1/sin θ • sec θ = 1/cos θ • cot θ = 1/tan θ • sin θ = 1/cosec θ • cos θ = 1/sec θ • tan θ = 1/cot θ Trigonometry Table: Below is the table for trigonometry formulas for angles that are commonly used for solving problems. Periodicity Identities (in Radians): These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities. • sin (π/2 – A) = cos A & cos (π/2 – A) = sin A • sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A • sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A • sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A • sin (π – A) = sin A & cos (π – A) = – cos A • sin (π + A) = – sin A & cos (π + A) = – cos A • sin (2π – A) = – sin A & cos (2π – A) = cos A Angles (In Degrees) 0 ° 30 ° 45 ° 60 ° 90 ° 18 0° 27 0° 36 0° Angles (In Radians) 0 π/6 π/4 π/3 π/ 2 π 3π/ 2 2π sin 0 1/2 1/ √2 √3/ 2 1 0 -1 0 cos 1 √3/ 2 1/ √2 1/2 0 -1 0 1 tan 0 1/ √3 1 √3 ∞ 0 ∞ 0 cot ∞ √3 1 1/ √3 0 ∞ 0 ∞ cosec ∞ 2 √2 2/ √3 1 ∞ -1 ∞ sec 1 2/ √3 √2 2 ∞ -1 ∞ 1 INVERSE TRIGNOMETRY FORMULAS: 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 Functions Domain Range Sin-1 x [-1, 1] [-π/2, π/2] Cos-1x [-1, 1] [0, π/2] Tan-1 x R (-π/2, π/2) Cosec-1 x R-(-1,1) [-π/2, π/2] Sec-1 x R-(-1,1) [0,π]-{ π/2} Cot-1 x R [-π/2, π/2]-{0} S.N o Inverse Trigonometric Formulas 1 sin-1(-x) = -sin-1(x), x ∈ [-1, 1] 2 cos-1(-x) = π -cos-1(x), x ∈ [-1, 1] 3 tan-1(-x) = -tan-1(x), x ∈ R 4 cosec-1(-x) = -cosec-1(x), |x| ≥ 1 5 sec-1(-x) = π -sec-1(x), |x| ≥ 1 6 cot-1(-x) = π – cot-1(x), x ∈ R 7 sin-1x + cos-1x = π/2 , x ∈ [-1, 1] 8 tan-1x + cot-1x = π/2 , x ∈ R 9 sec-1x + cosec-1x = π/2 ,|x| ≥ 1 10 sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1 11 cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1 12 tan-1(1/x) = cot-1(x), x > 0 13 tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1 14 tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1 15 2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1 16 2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0 17 2tan-1 x = tan-1(2x/(1-x2)), -1<x<1 18 3sin-1x = sin-1(3x-4x3) 19 3cos-1x = cos-1(4x3-3x) 20 3tan-1x = tan-1((3x-x3)/(1-3x2)) 21 sin(sin-1(x)) = x, -1≤ x ≤1 22 cos(cos-1(x)) = x, -1≤ x ≤1 23 tan(tan-1(x)) = x, – ∞ < x < ∞. 24 cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞ 25 sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞ 26 cot(cot-1(x)) = x, – ∞ < x < ∞. 27 sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2 28 cos-1(cos θ) = θ, 0 ≤ θ ≤ π 29 tan-1(tan θ) = θ, -π/2 < θ < π/2 30 cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2 31 sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π 32 cot-1(cot θ) = θ, 0 < θ < π 33 34 , if x, y ≥ 0 and x2+y2>1. 35 , if x, y ≥ 0 and x2+y2≤1. 36 , if x, y ≥ 0 and x2 +y2>1. 37 , if x, y >0 and x2+y2 ≤1. 38 , if x, y >0 and x2+y2>1. 39 , if x, y > 0 and x2+y2≤1.
