Trigonometry Examples

$\ln (\sqrt{1 - \sin (x)}) + \ln (\sqrt{1 + \sin (x)}) = \ln (\cos (\cos (x)))$

Step 1

Simplify $\ln (\sqrt{1 - \sin (x)}) + \ln (\sqrt{1 + \sin (x)})$ .

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\sqrt{1 - \sin (x)} \sqrt{1 + \sin (x)}) = \ln (\cos (\cos (x)))$

Step 1.2

Combine using the product rule for radicals.

$\ln (\sqrt{(1 - \sin (x)) (1 + \sin (x))}) = \ln (\cos (\cos (x)))$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 49.61759646, - 47.79566642, - 46.43470908, - 44.69318034, - 43.2539737, - 41.58863427, - 40.07550782, - 38.48191643, - 36.89942111, - 35.37292067, - 33.72581514, - 32.2615468, - 30.55478304, - 29.14770091, - 27.38640936, - 26.03129569, - 24.22076977, - 22.91225075, - 21.05793096, - 19.79049288, - 17.89795052, - 16.66595633, - 14.7408768, - 13.53858311, - 11.58674888, - 10.40832319, - 8.68569282, - 7.02227044, - 5.54410017, - 3.88067778, - 2.40250751, - 0.73908513, 0.73908513, 2.40250751, 3.88067778, 5.54410017, 7.02227044, 8.68569282, 10.40832319, 11.58674888, 13.53858311, 14.7408768, 16.66595633, 17.89795052, 19.79049288, 21.05793096, 22.91225075, 24.22076977, 26.03129569, 27.38640936, 29.14770091, 30.55478304, 32.2615468, 33.72581514, 35.37292067, 36.89942111, 38.48191643, 40.07550782, 41.58863427, 43.2539737, 44.69318034, 46.43470908, 47.79566642, 49.61759646$

Step 3