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  2. List of trigonometric identities - Wikipedia

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

    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.

  3. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes sin ⁡ ( 30 ∘ ) = 1 / 2 {\displaystyle \sin(30^{\circ ...

  4. Pythagorean trigonometric identity - Wikipedia

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

    In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity ...

  5. Trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Trigonometric_functions

    Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

  6. 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 ...

  7. Trigonometry - Wikipedia

    en.wikipedia.org/wiki/Trigonometry

    The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. [3] Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. [4]

  8. How to retire on less than $1 million and never run out of money

    www.aol.com/finance/retire-less-1-million-never...

    Bottom line. Ultimately, whether you can retire on less than $1 million will largely depend on your spending needs during retirement and your remaining life expectancy.

  9. 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 that is 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 ...