<p>Its in the Barrons guide, question 20 on the diagnostic test.</p>
<p>A) +- 0.55</p>
<p>C) 0.55</p>
<p>Why is the answer A, I am sure the solution is simple, and can you please not use CAST as an explanation.</p>
<p>This question has been driving me insane, I have tried solving using the cot^2x identity formula, and drawing out the triangle, and graphs but to no avail - please can someone explain this.</p>
<p>Thank you</p>
<p>You need to consider all the angles for which tan theta = 2/3. The first one is 33.69 degrees, but there are infinitely many, so find the next one and consider that.</p>
<p>@sylvan8798 TYBG, you are the first person to explain it and I understand it. However this seems kind of laborious to test for each angle. Is there a rule to remember that the angles always make ± answers or something like that.</p>
<p>Thanks though :)</p>
<p>Didn’t actually figure out the right explanation, unfortunately, but here’s what I was going to say before I realized my explanation was way off:</p>
<hr>
<p>Sorry for using CAST in my explanation here, but it’s all I can think of.</p>
<p>Tangent is positive in the first quadrant and the third quadrant. Triangle A (below) would have a tan theta value of 2/3, because (sin theta) and (cos theta) are both negative (meaning sin/cos will be positive), but so would Triangle B because (sin theta) and (cos theta) are both positive (meaning sin/cos will be positive).</p>
<p>TRIANGLE A (in the third quadrant)</p>
<p>[-3 units long]</p>
<hr>
<p>|’’’’’’’’’…–" [-2 units high]
|.–"</p>
<p>TRIANGLE B (in the first quadrant)</p>
<p>…<em>—"| [2 units high]
.–"".</em>__|
[3 units long]</p>
<p>So, if you know tan theta = 2/3, then either of those two triangles could be the case.</p>
<hr>
<p>…wait, never mind. Just realized that -0.55 radians would just go into the fourth quadrant, not the third. >_></p>
<p>Whatever, I’ll leave this here in case it leads to anything at all. Really hope someone can help you out, because now I’m just as curious about why -0.55 could be an answer as you are. :P</p>
<p>Uh -.55 is an answer if you add 180 degrees to the angle that gives .55. This will change the sign of sin and cos but without changing tan. BTW, you don’t need to calculate the angle like @sylvan8798 did. Just draw the triangle and notice that it’s similar to a 2,3,sqrt(13) right triangle.</p>
<p>@JohnSmith2014 @RandomHSer
I only get this from what the 1st user said, tan^-1(2/3)=33.7 so 33.7+180=213.7 so sin(213.7)= -0.55.</p>
<p>Thanks for your effort with CAST I kind of see where you are coming from. And also RandomHser that’s what I did when taking the test for the first time, but I missed the other angle, since I thought that either the 2 or the 3 had to be negative.</p>
<p>But yeah mean question,kind of get what has to be done now. Can anyone confirm whether what I did was right.</p>
<p>Alternatively:</p>
<p>Cot^2 x + 1 = cosec^2 x</p>
<p>since tan = 2/3 cot = 3/2</p>
<p>so (3/2)^2+1 = 1/sin^2
13/4 = 1/ sin^2
4/13 = sin^2 x</p>
<p>Now this leads to 2 roots. If you aren’t sure what I mean, do the substituion y = sin x</p>
<p>so 4/13 = y^2
This has 2 roots, one negative, one positive.</p>
<p>What you did was fine, although perhaps somewhat more time-consuming than you would want. It can be helpful to be able to sketch the tan, sin, and cos functions over several cycles (e.g. from -720 degrees to + 720 degrees) just conceptually. Then you are more able to see how they vary and how they relate.</p>
<p>@sylvan8798 Ah ok thanks, I think I know now though in future what to do for similar questions, and yeah thats a good point to consider too, thank you :)</p>
<p>You need to know what quadrant theta is in. Since tan theta > 0, theta is in the 1st or 3rd quadrant, but sin theta can equal 2/sqrt(13) or -2/sqrt(13). The test gives both numbers as choices, so without additional info, both can be correct.</p>
<p>@JohnSmith2014: I love those pixel triangles. So good.</p>