Solve the following trigonometric equations graphically for 0^ leq x leq 360^ : a) x = 0.71 b) x = 0.31

• Draw a coordinate plane with the x-axis from $0^\circ$ to $360^\circ$ and the y-axis from $-1$ to 1.
• Plot the key points for the sine function:
  • $(0^\circ, 0)$
  • $(90^\circ, 1)$ (maximum value)
  • $(180^\circ, 0)$
  • $(270^\circ, -1)$ (minimum value)
  • $(360^\circ, 0)$
• Connect these points with a smooth curve to form the sine wave.

To solve $\sin x = k$ using the graph, draw a horizontal line at $y = k$. The x-coordinates of the intersection points of this line with the sine curve are the solutions for $x$.

a) $\sin x = 0.71$
Draw a horizontal line at $y = 0.71$. Find the x-values where this line intersects the graph of $y = \sin x$.
The principal value is $x = \arcsin(0.71) \approx 45.2^\circ$.
The second solution in the range $0^\circ \le x \le 360^\circ$ is $x = 180^\circ - 45.2^\circ = 134.8^\circ$.
The solutions are approximately $\boxed{x = 45.2^\circ, 134.8^\circ}$.

b) $\sin x = 0.31$
Draw a horizontal line at $y = 0.31$. Find the x-values where this line intersects the graph of $y = \sin x$.
The principal value is $x = \arcsin(0.31) \approx 18.1^\circ$.
The second solution is $x = 180^\circ - 18.1^\circ = 161.9^\circ$.
The solutions are approximately $\boxed{x = 18.1^\circ, 161.9^\circ}$.

c) $\sin x = 0.6$
Draw a horizontal line at $y = 0.6$. Find the x-values where this line intersects the graph of $y = \sin x$.
The principal value is $x = \arcsin(0.6) \approx 36.9^\circ$.
The second solution is $x = 180^\circ - 36.9^\circ = 143.1^\circ$.
The solutions are approximately $\boxed{x = 36.9^\circ, 143.1^\circ}$.

d) $\sin x = -0.81$
Draw a horizontal line at $y = -0.81$. Find the x-values where this line intersects the graph of $y = \sin x$.
First, find the reference angle: $\alpha = \arcsin(0.81) \approx 54.1^\circ$.
Since $\sin x$ is negative, the solutions are in the third and fourth quadrants.
The solution in the third quadrant is $x = 180^\circ + 54.1^\circ = 234.1^\circ$.
The solution in the fourth quadrant is $x = 360^\circ - 54.1^\circ = 305.9^\circ$.
The solutions are approximately $\boxed{x = 234.1^\circ, 305.9^\circ}$.