Suppose [math]\theta[/math] is an angle drawn in standard position. [color=#666666][b]Let [i]P[/i]([i]x[/i], [i]y[/i]) be any point in the coordinate plane[/b][/color] and let[color=#666666][b] [i]r[/i] = the distance from [i]P[/i] to the origin[/b][/color]. [br][br]Recall [math]sin\left(\theta\right)=\frac{y}{r}[/math] and [math]csc\left(\theta\right)=\frac{r}{y}[/math]. [br][br]Interact with the applet below for a minute or two. Then answer the questions that follow. [br][color=#666666][b](Be sure to move point [i]P[/i] to various locations!) [/b][/color][br][br]
Regardless of where [i][color=#666666][b]P[/b][/color][/i] lies, what is the relationship between the values of the ratios [math]sin\left(-\theta\right)[/math] and [math]sin\left(\theta\right)[/math]?
Regardless of where [i][color=#666666][b]P[/b][/color][/i] lies, what is the relationship between the values of the ratios [math]csc\left(-\theta\right)[/math] and [math]csc\left(\theta\right)[/math]?
What do these 2 observations imply about the sine and cosecant functions? (Click [url=https://www.geogebra.org/m/pb8Drtd5]here[/url] and/or [url=https://www.geogebra.org/m/GY9tNvfB]here[/url] for a hint!)