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  2. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.

  3. Pythagorean trigonometric identity - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_trigonometric...

    In the figure, the point P has a negative x-coordinate, and is appropriately given by x = cos θ, which is a negative number: cos θ = −cos(π−θ). Point P has a positive y-coordinate, and sin θ = sin(π−θ) > 0.

  4. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. For real number x, the notation sin x, cos x, etc. refers to the value of the ...

  5. Sine and cosine - Wikipedia

    en.wikipedia.org/wiki/Sine_and_cosine

    In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...

  6. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...

  7. Chebyshev polynomials - Wikipedia

    en.wikipedia.org/wiki/Chebyshev_polynomials

    The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining ...

  8. cis (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Cis_(mathematics)

    x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.

  9. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians. [2] Since sin ⁡ ( x ) = cos ⁡ ( x − π / 2 ) , {\displaystyle \sin(x)=\cos(x-\pi /2),} the case of a sine can be omitted from this definition.