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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)

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