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The sign of the square root needs to be chosen properly—note that if 2 π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
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 ...
The half-angle formula for cosine can be obtained by replacing with / and taking the square-root of both sides: (/) = (+ ) /. Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle E B D {\displaystyle EBD} are all right-angled and similar, and all contain the angle θ ...
The tangents intercept at the radical line (in the diagram yellow). Similar considerations generate the second tangent circle, that meets the given circles at the points , (see diagram). All tangent circles to the given circles can be found by varying line . Positions of the centers
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial case, which is discussed below.
Trump's lawyers, citing presidential immunity and other ongoing litigation, told Merchan they oppose a hearing examining their claims of juror misconduct, and instead asked the judge to weigh the ...
Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have. In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this ...
Failed veep candidate Tim Walz is reportedly going to address the defeated Dems' 2024 donors Tuesday -- after going into obscurity in the weeks following his devastating loss with Vice President ...