The sine function, denoted as sin θ, is a trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions used in the study of polar coordinates and the representation of complex numbers in polar form.
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The sine function, sin θ, is one of the three primary trigonometric functions, along with cosine and tangent.
In a right triangle, sin θ is the ratio of the length of the opposite side to the length of the hypotenuse.
The sine function is periodic, with a period of $2\pi$, meaning that sin(θ + $2\pi$) = sin(θ).
The sine function is used to describe the $x$-coordinate of a point on the unit circle, where the angle θ is measured from the positive $x$-axis.
In polar coordinates, the sine function is used to determine the $y$-coordinate of a point, given the radius and angle.
Review Questions
Explain how the sine function, sin θ, is used in the context of polar coordinates.
In the polar coordinate system, the sine function, sin θ, is used to determine the $y$-coordinate of a point. The polar coordinates of a point are specified by a radius, $r$, and an angle, $θ$, measured counterclockwise from the positive $x$-axis. The $y$-coordinate of the point is then given by $r\sin θ$, where the sine function describes the ratio of the opposite side to the hypotenuse of the right triangle formed by the point and the polar coordinate axes.
Describe the relationship between the sine function, sin θ, and the representation of complex numbers in polar form.
The polar form of a complex number $z$ is given by $z = r(\cos θ + i\sin θ)$, where $r$ is the modulus (magnitude) of the complex number and $θ$ is the argument (angle) measured counterclockwise from the positive $x$-axis. The sine function, sin $θ$, is used to determine the imaginary component of the complex number in its polar form, as the $y$-coordinate of the point on the complex plane. This representation of complex numbers using the sine function is essential in various applications, such as the analysis of alternating current (AC) circuits.
Analyze the significance of the periodic nature of the sine function, sin θ, in the context of polar coordinates and the polar form of complex numbers.
The periodic nature of the sine function, with a period of $2\pi$, is crucial in the study of polar coordinates and the polar form of complex numbers. This periodicity means that the sine function repeats its values every $2\pi$ radians (or 360 degrees), which allows for the consistent and cyclical representation of points in the polar coordinate system and the angular component of complex numbers. The periodic nature of sin $θ$ ensures that complex numbers expressed in polar form can be easily transformed and manipulated, as the angle $θ$ can be adjusted by multiples of $2\pi$ without changing the underlying value of the complex number. This property is fundamental to many applications, such as the analysis of periodic phenomena in engineering and physics.
The fundamental trigonometric functions, including sine, cosine, and tangent, which describe the relationships between the sides and angles of a right triangle.
A coordinate system that specifies the location of a point in a plane by a distance from a reference point and an angle from a reference direction.
Polar Form of Complex Numbers: A way of representing complex numbers using a magnitude (modulus) and an angle (argument), rather than the standard Cartesian coordinates.