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We conclude that for 0 < θ < 1 / 2 π, the quantity sin(θ)/θ is always less than 1 and always greater than cos(θ). Thus, as θ gets closer to 0, sin(θ)/θ is "squeezed" between a ceiling at height 1 and a floor at height cos θ, which rises towards 1; hence sin(θ)/θ must tend to 1 as θ tends to 0 from the positive side:
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.
Differentiation under the integral sign – Differentiation under the integral sign formula; Hyperbolic functions – Collective name of 6 mathematical functions; Inverse hyperbolic functions – Mathematical functions; Inverse trigonometric functions – Inverse functions of sin, cos, tan, etc.
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 θ < θ. So we have < <. For negative values of θ we have, by the symmetry of the sine function
Euler's formula states that, for any real number x, one has = + , where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine").
The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations. The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge. Pictured is an accurate approximation of sin x around the point x = 0. The ...
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 ...
If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/ π)°, so that, for example, sin π = sin 180° when we take x = π.