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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.
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 involving the cotangent and the cosecant also follows from the Pythagorean theorem.
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 ...
Katharine Horgan's mom died when she was 7 years old. She left her jewelry and cards, which she treasured. She found the locket at a cafe in London.
Of the 283 misleading X posts that CCDH analyzed, 209, or 74% of the posts, did not show accurate notes to all X users correcting false and misleading claims about the elections, the report said.
Here are 10 performances that gave me great joy or moved me deeply—or both—in the last year. Every one is a tribute to what actors, at their best, can do—and a reason to keep going to the ...
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 ...
From January 2008 to June 2010, if you bought shares in companies when Allen U. Lenzmeier joined the board, and sold them when he left, you would have a -32.2 percent return on your investment, compared to a -26.9 percent return from the S&P 500.