Solution 4.3:4b
From Förberedande kurs i matematik 1
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- | {{ | + | If we once again use the Pythagorean identity we get |
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- | {{ | + | {{Displayed math||<math>\cos^2 v + \sin^2 v = 1\qquad\Leftrightarrow\qquad \sin v = \pm\sqrt{1-\cos^2 v}\,\textrm{.}</math>}} |
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+ | Because the angle ''v'' lies between <math>0</math> and <math>\pi</math>, <math>\sin v</math> is positive (an angle in the first and second quadrants has a positive ''y''-coordinate) and therefore | ||
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+ | {{Displayed math||<math>\sin v = +\sqrt{1-\cos^2 v} = \sqrt{1-b^2}\,\textrm{.}</math>}} |
Current revision
If we once again use the Pythagorean identity we get
\displaystyle \cos^2 v + \sin^2 v = 1\qquad\Leftrightarrow\qquad \sin v = \pm\sqrt{1-\cos^2 v}\,\textrm{.} |
Because the angle v lies between \displaystyle 0 and \displaystyle \pi, \displaystyle \sin v is positive (an angle in the first and second quadrants has a positive y-coordinate) and therefore
\displaystyle \sin v = +\sqrt{1-\cos^2 v} = \sqrt{1-b^2}\,\textrm{.} |