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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/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 \(y=x\).
