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sine
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{{other uses}}{{distinguish|sign|sign (mathematics)}}- the content below is remote from Wikipedia
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Right-angled triangle definition
(File:Trigono sine en2.svg|right|thumb| For the angle Î±, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.)To define the sine function of an acute angle Î±, start with a right triangle that contains an angle of measure Î±; in the accompanying figure, angle A in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:- The opposite side is the side opposite to the angle of interest, in this case side a.
- The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
- The adjacent side is the remaining side, in this case side b. It forms a side of (is adjacent to) both the angle of interest (angle A) and the right angle.
sin(alpha) = frac {textrm{opposite}} {textrm{hypotenuse}}
The other trigonometric functions of the angle can be defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides.As stated, the value sin(alpha) appears to depend on the choice of right triangle containing an angle of measure Î±. However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them.Unit circle definition
File:Unit circle.svg|right|thumb|Unit circle: the radius has length 1. The variable t measures the angleangleIn trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.Let a line through the origin, making an angle of Î¸ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to {{math|cos(Î¸)}} and {{math|sin(Î¸)}}, respectively. The point's distance from the origin is always 1.Unlike the definitions with the right triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.File:Circle_cos_sin.gif|thumb|450px|left|Animation showing how the sine function (in red) y = sin(theta) is graphed from the y-coordinate (red dot) of a point on the unit circleunit circle{{clr}}Identities
Exact identities (using radians):These apply for all values of theta.Reciprocal
The reciprocal of sine is cosecant, i.e., the reciprocal of {{math|sin(A)}} is {{math|csc(A)}}, or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:
csc(A) = frac{1}{sin(A)} = frac {textrm{hypotenuse}} {textrm{opposite}} = frac{h}{a}.
Inverse
(File:Arcsine.svg|thumb|180px|The usual principal values of the {{math|arcsin(x)}} function graphed on the cartesian plane. Arcsin is the inverse of sin.)The inverse function of sine is arcsine (arcsin or asin) or inverse sine ({{math|sin{{sup|-1}}}}). As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example, {{math|sin(0) {{=}} 0}}, but also {{math|sin({{pi}}) {{=}} 0}}, {{math|sin(2{{pi}}) {{=}} 0}} etc. It follows that the arcsine function is multivalued: {{math|arcsin(0) {{=}} 0}}, but also {{math|arcsin(0) {{=}} {{pi}}}}, {{math|arcsin(0) {{=}} 2{{pi}}}}, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression {{math|arcsin(x)}} will evaluate only to a single value, called its principal value.
theta = arcsin left( frac{text{opposite}}{text{hypotenuse}} right) = sin^{-1} left( frac {a}{h} right).
k is some integer:
begin{align}
sin(y) = x iff & y = arcsin(x) + 2pi k , text{ or }
& y = pi - arcsin(x) + 2pi k
end{align}Or in one equation:
sin(y) = x iff y = (-1)^k arcsin(x) + pi k
Arcsin satisfies:
sin(arcsin(x)) = x!
and
arcsin(sin(theta)) = thetaquadtext{for }-frac{pi}{2} leq theta leq frac{pi}{2}.
Calculus
{{See also|List of integrals of trigonometric functions|Differentiation of trigonometric functions}}For the sine function:
f(x) = sin(x)
The derivative is:
f'(x) = cos(x)
The antiderivative is:
int f(x),dx = -cos(x) + C
C denotes the constant of integration.Other trigonometric functions
(File:Sine cosine one period.svg|right|thumb|The sine and cosine functions are related in multiple ways. The two functions are out of phase by 90Â°: sin(pi/2 - x) = cos(x) for all angles x. Also, the derivative of the function {{math|sin(x)}} is {{math|cos(x)}}.)It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).Sine in terms of the other common trigonometric functions:{| class="wikitable"!colspan="1" rowspan="3"|!colspan="1" rowspan="3"|f Î¸!colspan="5"|Using plus/minus (Â±)!colspan="1"|Using sign function (sgn)!colspan="1" rowspan=2|f Î¸ =!colspan="4"|Â± per Quadrant!rowspan="2"|f Î¸ =! I! II! III! IV!rowspan="2"|cos|sin(theta)|= pmsqrt{1 - cos^2(theta)}| +| +| âˆ’| âˆ’|= sgnleft( cos left(theta - frac{pi}{2}right)right) sqrt{1 - cos^2(theta)}|cos(theta)|= pmsqrt{1 - sin^2(theta)}| +| âˆ’| âˆ’| +|= sgnleft( sin left(theta+ frac{pi}{2}right)right) sqrt{1 - sin^2(theta)}!rowspan="2"|cot| sin(theta)| = pmfrac{1}{sqrt{1 + cot^2(theta)}}| +| +| âˆ’| âˆ’| = sgnleft( cotleft( frac{theta}{2}right)right) frac{1}{sqrt{1 + cot^2(theta)}}|cot(theta)| = pmfrac{sqrt{1 - sin^2(theta)}}{sin(theta)}| +| âˆ’| âˆ’| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{sqrt{1 - sin^2(theta)}}{sin(theta)}!rowspan="2"|tan| sin(theta)| = pmfrac{tan(theta)}{sqrt{1 + tan^2(theta)}} | +| âˆ’| âˆ’| +| = sgnleft( tanleft(frac{2theta + pi}{4}right)right) frac{tan(theta)}{sqrt{1 + tan^2(theta)}}| tan(theta)| = pmfrac{sin(theta)}{sqrt{1 - sin^2(theta)}}| +| âˆ’| âˆ’| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{sin(theta)}{sqrt{1 - sin^2(theta)}}!rowspan="2"|sec| sin(theta)| = pmfrac{sqrt{sec^2(theta) - 1}}{sec(theta)} | +| âˆ’| +| âˆ’| = sgnleft( sec left( frac{4 theta - pi}{2}right)right) frac{sqrt{sec^2(theta) - 1}}{sec(theta)} | sec(theta)| = pmfrac{1}{sqrt{1 - sin^2(theta)}}| +| âˆ’| âˆ’| +| = sgnleft( sin left(theta+ frac{pi}{2}right)right) frac{1}{sqrt{1 - sin^2(theta)}}For all equations which use plus/minus (Â±), the result is positive for angles in the first quadrant.The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:
cos^2(theta) + sin^2(theta) = 1!
where sin2(x) means (sin(x))2.Sine squared function
(File:SinSquared.png|thumb|Sine function in blue and sine squared function in red. The Y axis is in radians.)The graph shows both the sine function and the b:Trigonometry/Graph of Sine Squared|sine squared]] function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.The sine squared function can be expressed as a modified sine wave from the Pythagorean identity and power reduction by the cosine double-angle formula:WEB, Sine-squared function,weblink August 9, 2019,
sin^2(theta) = frac{1 - sin(2theta+tfrac{pi}{2})}{2}
Properties relating to the quadrants
(File:Quadrants 01 Pengo.svg|thumb|200px|The four quadrants of a Cartesian coordinate system.)The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity sin(alpha + 360^circ) = sin(alpha) of the sine function.{| class="wikitable" style="text-align: center"!Quadrant!Degrees!Radians!Value!Sign!Monotony!Convexity|1st Quadrant|0^circ- content above as imported from Wikipedia
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