# Trigonometry

Right-angled triangle is a triangle in which one angle is a right angle, i.e. a 90-degree angle.

Hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. Two other sides are called legs of the triangle .

Let's choose any non-right angle α of the triangle, then

1. the sine of this angle is the ratio of the opposite side to the hypotenuse, denoted as sin(α)
2. the cosine of this angle is the ratio of the adjacent leg to the hypotenuse, denoted as cos(α)
3. the tangent of this angle is the ratio of the opposite leg to the adjacent leg, denoted as tg(α) or tan(α)

There are reciprocals of these functions:

1. cosecant: csc(α) = 1/cos(α)
2. secant: sec(α) = 1/cos(α)
3. cotangent: cot(α) = 1/tg(α)

## constants

The number π is a mathematical constant defined as the ratio of a circle's circumference to its diameter.

``π = 3.14159``

The number e or Euler's number is a mathematical constant defined as limit of (1 + 1/n)n where n approaches infinity.

``e = 2.71828``

The radian is the unit for measuring angles. Denoted by the symbol rad

An angle of one radian subtended from the center of a unit circle produces an arc with arc length 1. There is a formula for the correlation of degrees and radians:

``360° = 2 * π rad``

## basic formulas

``````sin2(α) + cos2(α) = 1

tan(α)*ctg(α) = 1

tan(α) = sin(α)/cos(α)=1/ctg(α)

1+tan2(α) = 1/cos2(α) = sec2(α)

ctg(α) = cos(α)/sin(α) = 1/tg(α)

1+ctg2(a) = 1/sin2(α) = cosec2(α)

sec(α)=1/cos(α)

cosec(α)=1/sin(α)
``````

## angle sum

``````sin(α±β) = sin(α)cos(β)±cos(α)sin(β)

cos(α+β) = cos(α)cos(β)-sin(α)sin(β)

cos(α-β) = cos(α)cos(β)+sin(α)sin(β)

tan(α+β) = (tan(α)+tan(β))/(1-tan(α)tan(β))
= (ctg(α)+ctg(β))/(ctg(α)ctg(β)-1)

tan(α-β) = (tan(α)-tan(β))/(1+tan(α)tan(β))
= (ctg(β)-ctg(α))/(ctg(α)ctg(β)+1)

ctg(α+β) = (ctg(α)ctg(β)-1)/(ctg(α)+ctg(β))
= (1-tan(α)tan(β))/(tan(α)+tan(β))

ctg(α-β) = (ctg(α)ctg(β)-1)/(ctg(β)-ctg(α))
= (1+tan(α)tan(β))/(tan(α)-tan(β))``````

## double angle

``````sin(2α) = 2sin(α)cos(α) = 2tan(α) / (1+tan2(α))

cos(2α) = cos2(α) - sin2(α)
= 2cos2(α) - 1
= 1 - 2sin2(α)
= (1 - tan2(α)) / (1+tan2(α))
= (ctg(α) - tan(α)) / (ctg(α) + tan(α))

tan(2α) = 2tan(α) / (1 - tan2(α))``````

There is the Chebyshev method for finding the nth multiple angle formula.

## signs in quarters

quater sin cos tgctgseccosec
I++++++
II+----+
III--++--
IV-+--+-

## parity and periods formulas

arg sin cos tanctgseccosec

2π-α
-sin(α) cos(α) -tan(α) -ctg(α)sec(α)-cosec(α)
0.5π-α
0.5π+α
cos(α) -sin(α)
sin(α)
-ctg(α)
ctg(α)
-tan(α)
tan(α)
-cosec(α)
cosec(α)
sec(α)
1.5π+α
1.5π-α
-cos(α)sin(α)
-sin(α)
-ctg(α)
ctg(α)
-tan(α)
tan(α)
cosec(α)
-cosec(α)
-sec(α)

## product to sum formulas

``````2cos(α)cos(β) = cos(α-β) + cos(α+β)

2sin(α)sin(β) = cos(α-β) + cos(α+β)

2sin(α)cos(β) = sin(α+β) + sin(α-β)

2cos(α)sin(β) = sin(α+β) - sin(α-β)

tan(α)tan(β) = ( cos(α-β) - cos(α+β)) / ( cos(α-β) + cos(α+β))
``````

## sum to product formulas

``````sin(α) + sin(β) = 2sin((α+β)/2)cos((α-β)/2)

sin(α) - sin(β) = 2sin((α-β)/2)cos((α+β)/2)

cos(α) + cos(β) = 2cos((α+β)/2)cos((α-β)/2)

cos(α) - cos(β) = -2sin((α+β)/2)cos((α-β)/2)
``````

## Euler's formula

In following formula i is imaginary unit.

``````ei*x = cos(x) + i*sin(x)

sin(x) = (eix - e-ix) / 2i

cos(x) = (eix + e-ix) / 2

tan(x) = (e-ix - eix) / (eix + e-ix)
``````