Trigonometric formulas

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1. Identities of the trigonometric functions of same the argument:

tangent function definition

cotangent function definition
basic trigonometric identity


tangent and cotangent connection
tangent and cosine connection

cotangent and sine connection





2. Values of the trigonometric functions at specific points:

α° 0 30 45 60 90 180 270 360
sin(α) 0 1/2 square root of 2 divide by 2 square root of 3 divide by 2 1 0 -1 0
cos(α) 1 square root of 3 divide by 2 square root of 2 divide by 2 1/2 0 -1 0 1
tg(α) 0 1 divide by square root of 3 1 square root of 3 0 0
ctg(α) square root of 3 1 1 divide by square root of 3 0 0

3. Evenness or oddness:

oddness of the sine function

oddness of the tangent function
evenness of the cosine function

oddness of the cotangent function



4. The signs at quarters:

Trig. function Quater
I II III IV
sin(α) + +
cos(α) + +
tg(α) + +
ctg(α) + +

5. Reduction formulas:

β = 90 ± α 180 ± α 270 ± α 360 ± α
sin(β) cos(α) ∓sin(α) cos(α) ±sin(α)
cos(β) ∓sin(α) cos(α) ±sin(α) cos(α)
tg(β) ∓ctg(α) ±tg(α) ∓ctg(α) ±tg(α)
ctg(β) ∓tg(α) ±ctg(α) ∓tg(α) ±tgctg(α)

6. Solution of the simplest trigonometric equations:

sin(x) = a ⇔ x = (-1)n arcsin(a) + n π, n = 0, 1 ... ∈ Z
cos(x) = a ⇔ x = ± arccos(a) + 2 n π, n = 0, 1 ... ∈ Z
tg(x) = a ⇔ x = arctg(a) + n π, n = 0, 1 ... ∈ Z

7. Sum and subtraction formulas:

the sine of the sum
the sine of the difference
the cosine of the sum
the cosine of the difference
the tangent of the sum
the tangent of the difference

8. Double-angle formulas:

sine of the double-angle
cosine of the double-angle
tangent of the double-angle

8. Triple-angle formulas:

sine of the triple-angle
cosine of the triple-angle
tangent of the triple-angle

9. Sum and difference transformation formulas:

a) Transformation of the sum and difference of the same trigonometric functions with different angles:

Sine transformations:

Sine transformations
sine of the difference

Cosine transformations:

cosine of the sum
cosine of the difference

Tangent transformations:

tangent of the sum
tangent of the difference

Cotangent transformations:

котангенс суммы
cotangent of the difference

b) Transformation of the sum and difference of the different trigonometric functions with different angles:

sum of the sine and cosine of the different angles
difference of the sine and cosine of the different angles

c) Specific formulas:

sine and cosine of different angles formula
common sine and cosine of different angles formula
sine sum up to n-th angle formula
cosine sum up to n-th angle formula

10. Trigonometric functions product transformation formulas:

sines of different angles product formula
sine and cosine of the different angles product formula
cosines of different angles product formula

11. Reduce trigonometric functions power formulas:

sine square transformation formula
cosine square transformation formula
sine square transformation formula
cosine square transformation formula

11. Expression of the trigonometric functions by the tangent of the half angle:

assume t is the tangent of the half angle
expression of the sine by tangent of half angle
expression of the cosine by tangent of half angle
expression of the tangent by tangent of half angle
expression of the cotangent by tangent of half angle
expression of the cotangent by tangent of half angle