Exercise 72 Page 586

Since $A$ is an obtuse angle, the rotation will be in Quadrant II. Furthermore, as it has a sine of $\frac{3}{10},$ the rotation should coincide with $\sin \theta =0.3$ on the unit circle.

Note that the radius is $1$ and the height of the plotted point is $0.3.$ We can add this information to our diagram. Let's also draw a right triangle.

In a right triangle, the tangent of an angle is the ratio of its opposite side to its adjacent side. Let's find the length of the second leg with the Pythagorean Theorem. The length of the second leg is $0.95.$ However, notice that the triangle is on the negative side of the horizontal axis. Therefore, the first coordinate of the point, which corresponds to $\cos A,$ is $\text{-}0.95.$ Recall that $\tan A$ is defined as $\frac{\sin A}{\cos A}.$ We already know that $\sin A=0.3,$ and we see above that $\cos A=\text{-}0.95.$ The tangent of $A$ is approximately $\text{-}0.32.$