91Ó°ÊÓ

: Step 1: Draw a right triangle To evaluate a trigonometric expression with an inverse function, such as this, we can begin by drawing a right triangle. Let the angle \(A\) be the angle whose sine is \(\frac{1}{5}\). Then, we have \(\sin A = \frac{1}{5}\) and the opposite side to angle \(A\) has length \(1\) and the hypotenuse has length \(5\). Step 2: Find the adjacent side By using Pythagorean theorem (\(a^2 + b^2 = c^2\)), we can find the length of the adjacent side. Let us denote the length of the adjacent side as \(b\): \[b^2 + 1^2 = 5^2\] \[b^2 = 24\] \[b = \sqrt{24}\] Step 3: Find the cosine Now that we know the lengths of all sides of the triangle, we can find the cosine of angle \(A\): \[\cos A = \frac{adjacent}{hypotenuse} = \frac{\sqrt{24}}{5}\] So \(\cos \left(\arcsin \left(\frac{1}{5}\right)\right) = \frac{\sqrt{24}}{5}\). ----------------------------------------------------------------------------------------------------------

: To determine \(\tan \left(\cos^{-1} \left(\frac{2}{3}\right)\right)\), we can follow a similar procedure as in part (a). Step 1: Draw a right triangle Let angle \(B\) be the angle whose cosine is \(\frac{2}{3}\). Then, we have \(\cos B = \frac{2}{3}\) and the adjacent side to angle \(B\) has length \(2\) and the hypotenuse has length \(3\). Step 2: Find the opposite side By using Pythagorean theorem, we can find the length of the opposite side. Let us denote the length of the opposite side as \(a\): \[a^2 + 2^2 = 3^2\] \[a^2 = 5\] \[a = \sqrt{5}\] Step 3: Find the tangent Now that we have the lengths of all sides, we can find the tangent of angle \(B\): \[\tan B = \frac{opposite}{adjacent} = \frac{\sqrt{5}}{2}\] So \(\tan \left(\cos^{-1} \left(\frac{2}{3}\right)\right) = \frac{\sqrt{5}}{2}\). ----------------------------------------------------------------------------------------------------------

: To determine \(\sin \left(\tan^{-1}(2)\right)\), we can follow a similar procedure as before. Step 1: Draw a right triangle Let angle \(C\) be the angle whose tangent is \(2\). Then, we have \(\tan C = 2\) and the opposite side to angle \(C\) has length \(2\) and the adjacent side has length \(1\). Step 2: Find the hypotenuse By using Pythagorean theorem, we can find the length of the hypotenuse. Let us denote the length of the hypotenuse as \(c\): \[1^2 + 2^2 = c^2\] \[c^2 = 5\] \[c = \sqrt{5}\] Step 3: Find the sine Now that we know the lengths of all sides of the triangle, we can find the sine of angle \(C\): \[\sin C =\frac{opposite}{hypotenuse} = \frac{2}{\sqrt{5}}\] So \(\sin \left(\tan^{-1}(2)\right) = \frac{2}{\sqrt{5}}\). ----------------------------------------------------------------------------------------------------------

: Since \(\arcsin(-\frac{1}{5})\) corresponds to a negative angle, we are dealing with an angle in the fourth quadrant. The cosine is positive in the fourth quadrant; thus, \(\cos(\arcsin(-\frac{1}{5})) = \cos(\arcsin(\frac{1}{5}))\): \(\cos \left(\arcsin \left(-\frac{1}{5}\right)\right) = \frac{\sqrt{24}}{5}\) ----------------------------------------------------------------------------------------------------------

: To determine \(\sin \left(\arccos \left(-\frac{3}{5}\right)\right)\), we can follow a similar procedure as before. Step 1: Draw a right triangle Let angle \(E\) be the angle whose cosine is \(-\frac{3}{5}\). Then, we have \(\cos E = -\frac{3}{5}\) and the adjacent side to angle \(E\) has length \(-3\) and the hypotenuse has length \(5\). Step 2: Find the opposite side By using Pythagorean theorem, we can find the length of the opposite side. Let us denote the length of the opposite side as \(a\): \[a^2 + (-3)^2 = 5^2\] \[a^2 = 16\] \[a = 4\] Step 3: Find the sine Now that we know the lengths of all sides of the triangle, we can find the sine of angle \(E\): \[\sin E = \frac{opposite}{hypotenuse} = \frac{4}{5}\] So \(\sin \left(\arccos \left(-\frac{3}{5}\right)\right) = \frac{4}{5}\). These are the exact values of each expression. You can now check the results with a calculator.