The Unit Circle (can't really explain any more, sorry)

[Freddie2]

I have a pretty basic and straightforward question (I think). In the example below, it asks to solve the equations to 1 decimal place, pretty straight forward. However on the last one, it states that there are two possible answers? I understand how they got the two answers, but I'm wondering, how do you know when to go for two answers instead of one. How come you don't do it for the other ones, but you do it for the last one? Thanks!

 

[stapel]

In the example below...on the last one, it states that there are two possible answers? I understand how they got the two answers, but I'm wondering, how do you know when to go for two answers instead of one. How come you don't do it for the other ones, but you do it for the last one?
View attachment 7936

You "know" by looking at the pictures they've given you. Since "zero to one eighty" is "halfway around" or "the first two quadrants", you have to look for values in that location. There was only one place where cosine (being the "x" distance in the unit circle) was positive, and similarly only one place where the cosine was negative. However, since the sine (being the "y" distance) is positive in both the first and second quadrant, there were two possible solutions to the equation. (There would have been no solutions, had they given you a negative value for the sine.)

Learn the graphs and pictures well. You'll need them for more than just this exercise!

 

[ksdhart2]

Another way you might think about it to graph the functions involved. For instance, in the first one, graph $\displaystyle y=cos(x)$ and $\displaystyle y=0.623$ for $\displaystyle 0^{\circ} \le x \le 180^{\circ}$ (or, in radians, $\displaystyle 0 \le x \le \pi$). There you'll see that the two functions intersect in exactly one point, meaning there's just one solution to $\displaystyle cos(x)=0.623$ in the specified domain. Similarly, graphing $\displaystyle cos(x)$ and $\displaystyle y=-0.317$ will reveal just one solution in the specified domain. However, $\displaystyle sin(x)$ and $\displaystyle y=0.814$ intersect twice in the given domain. Per Stapel's note, if the equation had been $\displaystyle sin(x)=-0.814$, you'd see the functions don't intersect at all in the given domain, meaning there's no solutions.