0

Suppose I'm asked to solve for $\cos\theta=−0.5$, for $\theta$ between $0^\circ$ and $360^\circ$ inclusive.

I am told that the first step would be to buy the basic angle, $\alpha$. The way I am told to do this is by omitting the negative sign from -0.5 and hence finding sine inverse of 0.5 to get $60^\circ$.

I understand the next steps of identifying the quadrant $\theta$ lies in etc: what I don't understand is why I need to omit the negative sign when finding the basic angle.

Aurora
  • 3
  • 1
    You don't explicitly have to, however it is common to do. There is a large amount of symmetry going on here. All you are expected to memorize in terms of standard common angles are those angles in the first quadrant and their respective sine and cosine values. All other angles in the other quadrants and their sine and cosine values are exactly the same, just reflected. – JMoravitz Feb 23 '21 at 13:25
  • $\alpha$ is an angle you know is between $0^o$ and $90^o$. The minus sign on the cos tells you the angle $\theta$ you want is in 2nd or 3rd quadrant. In the second quadrant $\theta = 180-\alpha$ and in the third quadrant $\theta = 180+\alpha$ (measured anti-clockwise from the positive horizontal axis). This approach will work for sin, cos and tan calculations. – Paul Feb 23 '21 at 13:33
  • Why would you exactly duplicate (most of) a question from four years ago? – Blue Feb 23 '21 at 14:00
  • @Blue interesting find. Very strange... This post contains the exact same bizarre grammatical mistakes as well, practically confirming that it was copy-pasted. "...would be to buy the basic..." for example. Huh... – JMoravitz Feb 23 '21 at 17:35
  • @Blue Umm..I didn’t get the answer of that question and his question is exactly what i wanted to ask, so i just wanna post that question again :) – Aurora Feb 24 '21 at 01:14
  • @Aurora: "I didn’t get the answer of that question [...] so i just wanna post that question again" ... Simply reposting someone else's question isn't the best idea. Better: "I found this old question. [Quote it, and link to it.] I didn't quite understand the answer given. [Explain what you found unsatisfactory.]" The second bit is especially important, as it helps answerers target the specific source of your trouble, without wasting time repeating confusing notions from the previous answer. Generally speaking, the more you can tell us about your thoughts on a problem, the better. Cheers :) – Blue Feb 24 '21 at 06:28
  • @blue Noted, thank you for the suggestion! – Aurora Feb 24 '21 at 13:17

1 Answers1

3

What you are searching for is the reference angle $\theta$ which is in the first quadrant. Once we have that, then the candidate angles are $\theta, 180-\theta, 180+\theta,$ and $360-\theta$. But since the trig functions for angles in the first quadrant are always positive, we want to consider the absolute value of the ratio.

There is also probably a historical artifact. Before scientific calculators, one would have to use trig tables to calculate inverse trig values, and those would only contain positive ratios and angles from $0$ to $90$. Nowadays, it is much more natural to find at least one solution for $\cos\theta=-0.5$ using a scientific calculator, but it probably still makes sense to find the reference angle in the first quadrant so that you can find the other solution without too much drama.