Hey Guys,
I just had a question regarding the Cambridge Specialist Maths Worked Example.
Can anyone please explain this to me, as attached in the image below.
Thanks
I believe that this question is focusing on the symmetry property in the unit circle which means that there are two quadrants where sin is positive (1st and 2nd) and where sin is negative (3rd and 4th).
So firstly, from inspection of the graph, it can be seen that
a would be in the first quadrant as it is before the turning point and it has a positive y coordinate. So from here, the question asks for all points from between [0, 2π], so you want to find all the other points around the unit circle for when
a would be positive which would be in the 2nd quadrant. So to get to the 2nd quadrant, you would need a reference angle which is
a. From there, you would need to do π -
a (based on the symmetry property), thus providing the other solution for between [0, 2π].
Secondly, upon inspection of the graph, it can be seen that
b is in the 2nd quadrant as it's positive and after the turning point. So for the other solution, you would want to get an answer in the first quadrant. since
b is in the 2nd quadrant, you can just use
b as the reference angle and do π -
b to get the angle symmetric to
b in the first quadrant.
Thirdly, for
c, it can be inspected that
c is negative but before the minimum turning point, therefore, it would be in the 3rd quadrant. So you would be looking for another answer in the 4th quadrant as sin is also negative in the 4th quadrant. To get an answer in the 4th quadrant, you would need to get a reference angle (the base angle). The answer gets this by doing
c - π. Then based on the symmetry property, to get an answer in the 4th quadrant, you would do 2π - x, for this solution it would be 2π - (
c - π) and that simplifies to 3π -
c.
Lastly, for x =
d, this is in the 4th quadrant as it's negative and past the minimum turning point. Just like for
c, you would need to find a solution in the 3rd quadrant. So firstly, you would find a reference angle, which would be 2π -
d. Then to get to the 3rd quadrant, you would do π + (2π -
d), based on the symmetry property. Thus, this would simplify to 3π -
d.
I believe that's how they got all the answers for the values, so overall, it's just based on the symmetry property.