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Numbers, trigonometric functions and coordinate geometry
Published in Alan Jeffrey, Mathematics, 2004
The fundamental trigonometric functions are the sine, cosine and tangent functions. These are most conveniently defined in terms of the coordinates of a representative point P on a circle of unit radius, called a unit circle, centred on the origin. Consider the unit circle in Fig. 1.6 with the point P(x, y) located on its circumference, and the line OP making an angle θ that is measured anticlockwise from the x-axis, as shown. Then, for 0 ≤ θ ≤2π, we will define the sine, cosine and tangent functions as follows: the sine of θ, written sin θ, is the y-coordinate of point P,the cosine of θ, written cos θ, is the x-coordinate of point P,the tangent of θ, written tan θ, is given by tanθ=sinθcosθ.
Kinematics in Linear Motion
Published in Emeric Arus, Biomechanics of Human Motion, 2017
A trigonometric equation is an equation consisting of trigonometric functions. Trigonometric identity expressing a trigonometric equation that becomes true when the variable is replaced by every permissible number. Trigonometric ratio is a ratio that describes a relationship between a side and an angle of a triangle. Inverse trigonometric functions are the inverse functions of the trigonometric functions, written sin−1 (arcsin), cos−1 (arccos), tan−1 (arctan), cot−1 (arccot), sec−1 (arcsec), and csc−1 (arccsc). Here, the author will describe the trigonometric functions using the triangle shown by Figure 6.9.
Geometry and trigonometry revision for a.c.
Published in Adrian Waygood, An Introduction to Electrical Science, 2013
Trigonometry is the branch of mathematics which is concerned about the relationship between the sides and angles of triangles. In this chapter, we are only concerned with right-angled triangles because, as we shall see, when we deal with phasors (in single-phase a.c. circuits, at least) we treat them as right-angled triangles.
Quantum-inspired ant colony optimisation algorithm for a two-stage permutation flow shop with batch processing machines
Published in International Journal of Production Research, 2020
Zhen Chen, Xu Zheng, Shengchao Zhou, Chuang Liu, Huaping Chen
The number of parameters in the proposed algorithm is less than that in the general ant colony algorithm; the parameters include the population size, growth rate and number of iterations. In the comparison with BHACO (Zheng, Zhou, and Chen 2018) and HDDE (Chen et al. 2014), we set the population size to 30 and the number of iterations to 500. For the trigonometric function , the period is π. Assuming that a feature is enhanced in every iteration, the increment is after 500 iterations. The initial value is 1/2, which increases to 1, drops to 0, and increases to 1, in cycles. Therefore, when the amplitude is very small, the growth rate of a single iteration will be limited to the local optimum, and when the amplitude is too large, the original process will often be repeated.