• The solution to Newton's GEXACT VALUE & FORMULA
  • The theory controlling planetary spinTHE MATHEMATICAL LAW
  • The pressure at the centre of a massEARTH'S CORE PRESSURE (calculation procedure)
  • Proof of the non-exitence of Dark MatterDOES NOT EXIST
  • The atom as Newton and Coulomb describe itNO NEED FOR A UNIFICATION THEORY
The solution to Newton's G1 The theory controlling planetary spin2 Pressure at the centre of the Earth3 Proof of the non-exitence of Dark Matter4 The atom as Newton describes it5
Useful Stuff Algebra Trig Functions Calculus Max-Min Differentiation Rules Differentiation Trig Differentiation Logs Integration Methods Standard Integrals Stiffness & Capacity Mohr's Circle Earth's Atmosphere Earth's Properties Stars & Planets Laws of Motion Solar System Orbits Planetary Spin Rydberg Atom Planck Atom Newton Atom The Atom Dark Matter? Core Pressure The Big Bang Brakes and Tyres Vehicle Impacts Speeding vs Safety Surface Finish Pressure Classes Hardness Conversion Thermodynamics Steam (properties) Heat Capacity Work Energy Power Constants

Alternative Expressions for Compound Trigonometric Functions

Sin = O/H & Csc = 1/Sin
Cos = A/H & Sec = 1/Cos
Tan = O/A & Cot = 1/Tan
Where: O = opposite, A = adjacent & H = Hypotenuse

The following table contains equivalent or alternative ways to express trigonometric compound formulas.

Sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + x⁹/9! - x¹¹/11! + x¹³/13! - x¹⁵/15! +… etc.

Cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + x⁸/8! - x¹⁰/10! + x¹²/12! - x¹⁴/14! +… etc.

Sinh(x) = x¹/1 + x³/3 + x⁵/5 + x⁷/7 + x⁹/9 + x¹¹/11 + x¹³/13 + ....

Cosh(x) = 1 + x²/2 + x⁴/4 + x⁶/6 + x⁸/8 + x¹⁰/10 + x¹²/12 + x¹⁴/14 + ....

Cosine Rule: A² = B² + C² – 2.B.C.Cos(a)

Sine Rule: A/Sin(a) = B/Sin(b) = C/Sin(c)

Cos²(x) = 1 – Sin²(x)

Sin²(x) = 1 – Cos²(x)

Tan²(x) = Sec²(x) - 1

Cos²(x) + Sin²(x) = 1

Tan(π/3) = √3

Cos(2x) = Cos²(x) – Sin²(x) = 1 – 2.Sin²(x) = 2.Cos²(x) – 1

Sin(x/2) = ±(½(1 – Cos(x)))½

Sin(x) = 2.Sin(x/2).Cos(x/2)

Sin(2x) = 2.Cos(x).Sin(x)

½Sin(2x) = Sin(x).Cos(x)

Tan(x/2) = Sin(x)/(1+Cos(x)) = (1-Cos(x))/Sin(x)

Tan(x) = Sin(x) / Cos(x)

Tan²(x) = Sin²(x) / Cos²(x)

Tan(2x) = 2.Tan(x) / (1 – Tan²(x))

Cos(x/2) = ±(½(1 + Cos(x)))½

Cos(x) = (1 – Sin²(x))½

Sin(x) = (1 – Cos²(x))½

Tan(x) = Sin(x) / (1 – Sin²(x))½

1/Cos²(x) = 1 + tan²(x)

Sin(x).Cos(x) = ½.Sin(2.x)

Sin(x).Sin(y) = ½(Cos(x-y) - Cos(x+y))

Sin(x).Cos(y) = ½(Sin(x+y) + Sin(x-y))

Cos(x).Sin(y) = ½(Sin(x+y) - Sin(x-y))

Cos(x).Cos(y) = ½(Cos(x+y) + Cos(x-y))

Sin(x + y) = Sin(x).Cos(y) + Cos(x).Sin(y)

Cos(x + y) = Cos(x).Cos(y) – Sin(x).Sin(y)

Tan(x + y) = Tan(x)+Tan(y) / (1 – Tan(x).Tan(y))

Sin(x – y) = Sin(x).Cos(y) – Cos(x).Sin(y)

Cos(x – y) = Cos(x).Cos(y) + Sin(x).Sin(y)

Tan(x – y) = Tan(x) – Tan(y) / (1 + Tan(x).Tan(y))

Sin(x) + Sin(y) = 2.Sin(½(x + y)).Cos(½(x – y))

Cos(x) + Cos(y) = 2.Cos(½(x + y)).Cos(½(x – y))

Tan(x) + Tan(y) = Sin(x + y) / Cos(x).Cos(y)

Sin(x) – Sin(y) = 2.Cos(½(x + y)).Sin(½(x – y))

Cos(x) – Cos(y) = –2.Sin(½(x + y)).Sin(½(x – y))

Tan(x) - Tan(y) = Sin(x – y) / Cos(x).Cos(y)

Sin²(x) + Cos²(x) = 1

Tan(x) = (Sec²(x) – 1)½

Cot(x) = (Csc²(x) – 1)½

Sin(Acos(x)) = Cos(Asin(x)) = (1 – x²)½

Cos(Atan(x)) = 1 / (1 + x²)½

Tan(Acos(x)) = (1 – x²)½ / x                

Sin(Atan(x)) = x / (1 + x²)½

Tan(Asin(x)) = x / (1 – x²)½

Cosh(x) = ½(ex + e-x)

Cosh(x) + Sinh(x) = ex

Cosh²(x) – Sinh²(x) = 1

Sinh(x/2) = ±(½(Cosh(x)-1))½

Sinh(x) = ½(ex – e-x)

Cosh(x/2) = (½(Cosh(x)+1))½

Cosh(x) – Sinh(x) = e-x

Cosh²(x) + Sinh²(x) = Cosh(2x)

Tanh(x/2) = (Cosh(x)-1)/Sinh(x) = Sinh(x)/(Cosh(x)+1)

Tanh(x) = Sinh(x) / Cosh(x) = (ex – e-x)/(ex + e-x) = (e2x – 1)/(e2x + 1)

Csch(x) = 1/Sinh(x) = 2 / (ex – e-x)

Sech(x) = 1/Cosh(x) = 2 / (ex + e-x)

Coth(x) = (ex + e-x) / (ex – e-x)

Asinh(x) = logₑ(x + (x² + 1)½)

Acosh(x) = logₑ(x + (x² – 1)½)

Atanh(x) = ½ Logₑ((1 + x) / (1 – x))

Asinh(x/a) = logₑ((x + (x² + a²)½) / a)

Acosh(x/a) = logₑ(x + (x² – a²)½) / a)

Atanh(x/a) = ½ Logₑ((a + x) / (a – x))

3-D Trigonometric Functions and Relationships

3D Angles and Lengths

3D trigonometric functions and relationships

3D Angles and Lengths

R² = x² + y² + z²

Sin²(α) / Tan²(ψ) = 1 / Sin²(β) - 1

Direction Cosines

Direction cosines of a 3D scalar vector

Direction Cosines (l, m and n)

l = Cos(θx ) = P/x

m = Cos(θy ) = P/y

n = Cos(θz ) = P/z

Further Reading

You will find further reading on this subject in reference publications(19)

      Go to our store
CalQlata™ Copyright ©2011-2017 CalQlata info@calqlata.com Site Map Terms of website use Our Store