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- https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/04%3A_Inverse_Trigonometric_Functions/4.03%3A_Inverse_Trigonometric_PropertiesIf \(\theta\) is not in this domain, then we need to find another angle that has the same cosine as \(\theta\) and does belong to the restricted domain; we then subtract this angle from \(\dfrac{\pi}{...If \(\theta\) is not in this domain, then we need to find another angle that has the same cosine as \(\theta\) and does belong to the restricted domain; we then subtract this angle from \(\dfrac{\pi}{2}\).Similarly, \(\sin \theta=\dfrac{a}{c}=\cos\left(\dfrac{\pi}{2}−\theta\right)\), so \({\cos}^{−1}(\sin \theta)=\dfrac{\pi}{2}−\theta\) if \(−\dfrac{\pi}{2}≤\theta≤\dfrac{\pi}{2}\).
- https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/04%3A_Inverse_Trigonometric_Functions/4.01%3A_Basic_Inverse_Trigonometric_FunctionsThis equation is correct ifx x belongs to the restricted domain\(\left[−\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), but sine is defined for all real input values, and for \(x\) outside the restricted inte...This equation is correct ifx x belongs to the restricted domain\(\left[−\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), but sine is defined for all real input values, and for \(x\) outside the restricted interval, the equation is not correct because its inverse always returns a value in \(\left[ −\dfrac{\pi}{2},\dfrac{\pi}{2} \right]\). The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/06%3A_Analytic_Trigonometry/6.01%3A_Inverse_Trigonometric_FunctionsIn this section, we will explore inverse trigonometric functions. An inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and...In this section, we will explore inverse trigonometric functions. An inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa.
- https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/04%3A_Inverse_Trigonometric_Functions/4.02%3A_Graphing_Inverse_Trigonometric_FunctionsWe see that \({\sin}^{−1}x\) has domain \([ −1,1 ]\) and range \(\left[ −\dfrac{\pi}{2},\dfrac{\pi}{2} \right]\), \({\cos}^{−1}x\) has domain \([ −1,1 ]\) and range \([0,\pi]\), and \({\tan}^{−1}x\) h...We see that \({\sin}^{−1}x\) has domain \([ −1,1 ]\) and range \(\left[ −\dfrac{\pi}{2},\dfrac{\pi}{2} \right]\), \({\cos}^{−1}x\) has domain \([ −1,1 ]\) and range \([0,\pi]\), and \({\tan}^{−1}x\) has domain of all real numbers and range \(\left(−\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\). Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line \(y=x\).
