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Trigonometric functions

lastmod 2025-01-17created 2014-09-1112 edits

Trigonometric functions are functions of an angle used in mathematics.

Trigonometric functions are mainly used in assumptions about triangles or periodic phenomena. They are generally defined as the ratio of two sides of a right triangle containing the angle, and are sometimes defined as ratios of variable arc lengths on the unit circle. They can also be expressed as infinite series or as solutions of certain differential equations, which extends their domain to arbitrary positive and negative values, or to complex numbers. There are 6 basic trigonometric functions.

Categories

Basic knowledge of trigonometric function notation

When an exponent is written on a trigonometric function, the following holds.

  • (sin(θ))² = sin²(θ) = sin²θ
  • sin²θ ≠ sin(θ²)

Right triangle

Geometric definition

For the right triangle in the figure on the right(a: height, b: base, h: hypotenuse)1, the following formulas hold.

  • Sine: sin A = a / h
  • Cosine: cos A = b / h
  • Tangent: tan A = a / b = sin A / cos A
  • Cosecant: csc A = 1 / sin A
  • Secant: sec A = a / cos A
  • Cotangent: cot A = 1 / tan A

Trigonometric ratios of special angles

(ratios using a right triangle) In the figure on the right, the special angles below correspond to A, and the side ratio is b:a:h.

Special angle (angle between base and hypotenuse)

30º

45º

60º

Side ratio (base:height:hypotenuse)

\(\sqrt{3} : 1 : 2\)

\(1 : 1 : \sqrt{2}\)

\(1 : \sqrt{3} : 2\)

sin(θ)

\(1 / 2\)

\(1 / \sqrt{2}\)

\(\sqrt{3} / 2\)

cos(θ)

\(\sqrt{3} / 2\)

\(1 / \sqrt{2}\)

\(1 / 2\)

tan(θ)

\(1 / \sqrt{3}\)

\(1\)

\(\sqrt{3}\)

Triangle formula calculation table

Known angles or sides

Angles or sides to find

h, A || b = h × cos A || a = h × sin A

b, A || a = b × tan A || h = b / cos A

b, a || A = cot( a / b) || h² = b² + a²

b, h || a² = h² - b² || A = sec(b + h)

a, A || b = a / tan A || h = a / sin A

a, h || A = csc(a / h) || b² = h² - a²

Relationships between trigonometric ratios

Trigonometric ratios of acute angles

  • sin(90º - A) = cos A
  • cos(90º - A) = sin A
  • tan(90º - A) = cot A

Trigonometric ratios of supplementary angles

  • sin(180º - A) = sin A
  • cos(180º - A) = - cos A
  • tan(180º - A) = - tan A

Trigonometric ratios of the same angle

  • sin²A + cos²A = 1
  • tan A = sin A / cos A
  • a + tan²A = 1 / cos²A

Theorems of trigonometric functions

Addition theorem 

sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)
sin(x - y) = sin(x) * cos(y) - cos(x) * sin(y)
cos(x + y) = cos(x) * cos(y) - sin(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))
tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x) * tan(y))

Double-angle theorem 

sin(2 * x) = 2 * sin(x) * cos(x)
cos(2 * x) = cos²(x) - sin²(x) = 2 * cos²(x) - 1 = 1 - 2 * sin²(x)
tan(2 * x) = (2 * tan(x)) / (1 - tan²(x))

Half-angle theorem 

sin²(x / 2) = (1 - cos(x)) / 2
cos²(x / 2) = (1 + cos(x)) / 2
tan²(x / 2) = (1 - cos(x)) / (1 + cos(x))

Complete summary of trigonometric formulas 

sin(-θ)=-sinθ, cos(-θ)=cosθ, tan(-θ)=-tanθ
sin²θ+cos²θ=1, sec²θ-tan²θ=1, csc²θ-cot²θ=1
sin(π/2-θ)=cosθ, sin(π/2+θ)=cosθ, sin(θ±π/2)=±cosθ
cos(π/2-θ)=sinθ, cos(π/2+θ)=-sinθ, cos(θ±π/2)=Ŧsinθ
sin(π-θ)=sinθ, sin(π+θ)=-sinθ, sin(θ±π)=-sinθ
cos(π-θ)=-cosθ, cos(π+θ)=-cosθ, cos(θ±π)=-cosθ
sin(α±β)=sinαcosβ±cosαsinβ, cos(α±β)=cosαcosβŦsinαsinβ
tan(α+β)=tanα+tanβ/1-tanαtanβ, tan(α-β)=tanα-tanβ/1+tanαtanβ
sin2θ=2sinθcosθ
cos2θ=cos²θ-sin²θ=2cos²θ-1=1-2sin²θ, tan2θ=2tanθ/1-tan²θ
sin²θ/2=1-cosθ/2, cos²θ/2=1+cosθ/2, tan²θ/2=1-cosθ/1+cosθ
sinαcosβ=1/2{sin(α+β)+sin(α-β)}
cosαcosβ=1/2{cos(α+β)+cos(α-β)}
sinαsinβ=-1/2{cos(α+β)-cos(α-β)}
sinα+sinβ=2sin(α+β/2)cos(α-β/2)
sinα-sinβ=2cos(α+β/2)sin(α-β/2)
cosα+cosβ=2cos(α+β/2)cos(α-β/2)
cosα-cosβ=-2sin(α+β/2)sin(α-β/2)

See also

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References

  1. Hypotenuse, Adjacent (base), Opposite (height) ↩

  2. Wikipedia-trigonometric_functions_ko.pdf ↩

  3. List_of_trigonometric_identities.pdf ↩

  4. List_of_trigonometric_identities_ko.pdf ↩

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