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  2. Pythagorean trigonometric identity - Wikipedia

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

    Point P has a positive y-coordinate, and sin θ = sin(π − θ) > 0. As θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant.

  3. List of trigonometric identities - Wikipedia

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

    Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them.

  4. 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 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 of the ...

  5. Mathematics and God - Wikipedia

    en.wikipedia.org/wiki/Mathematics_and_God

    The application is a defense of Christianity stating that "If God does not exist, the Atheist loses little by believing in him and gains little by not believing. If God does exist, the Atheist gains eternal life by believing and loses an infinite good by not believing". [3] The atheist's wager has been proposed as a counterargument to Pascal's ...

  6. History of trigonometry - Wikipedia

    en.wikipedia.org/wiki/History_of_trigonometry

    [27] [28] Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the angle , radius , diameter , and circumference of a circle in terms of trigonometric functions.

  7. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

  8. 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 ...

  9. Five Ways (Aquinas) - Wikipedia

    en.wikipedia.org/wiki/Five_Ways_(Aquinas)

    Ward defended the utility of the five ways (for instance, on the fourth argument he states that all possible smells must pre-exist in the mind of God, but that God, being by his nature non-physical, does not himself stink) whilst pointing out that they only constitute a proof of God if one first begins with a proposition that the universe can ...

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