Trigonometric formulas

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

tangent function definition
basic trigonometric identity
tangent and cosine connection
cotangent function definition
tangent and cotangent 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
evenness of the cosine function
oddness of the tangent 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) ∓ctg(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

9. Triple-angle formulas:

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

10. Sum and difference transformation formulas:

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

Sine transformations:

sum of sines
difference of the sines

Cosine transformations:

sum of cosines
difference of the cosines

Tangent transformations:

sum of tangents
difference of the tangents

Cotangent transformations:

sum of cotangents
difference of the cotangents

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:

sum/difference of sine and cosine of same angle formula
sine and cosine of same angle sum/difference common formula
sine sum up to n-th angle formula
cosine sum up to n-th angle formula

11. Trigonometric functions product transformation formulas:

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

12. Reduce trigonometric functions power formulas:

sine square reduction formula
cosine square reduction formula
sine cube reduction formula
cosine cube reduction formula

13. 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 tangent by tangent of half angle
expression of the cosine by tangent of half angle
expression of the cotangent by tangent of half angle

Other useful links:

Partial derivative step by step sample
Formulas of arithmetical and geometrical progressions
Logarithms identities

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