William W. answered • 02/17/23
Experienced Tutor and Retired Engineer
Start with cos(θ) = -0.8910 then take cos inverse of both sides:
cos-1[cos(θ)] = cos-1(-0.8910)
cos-1[cos(θ)] is just θ so you get:
θ = cos-1(-0.8910) which you can then just plug in your calculator. Use "degree" mode since they tell you your answer is to be between 90° and 180°. Your calculator should say θ = 152.9991766 or essentially 153°.
Now, you need to consider if this is the correct answer based on the restrictions given.
Given that your are restricted to θ between 90° and 180°, your answer must reflect cosine being in quadrant II. So we could sketch this situation like this:
You can now compare this to the answer you got and it checks out. 153° would match that sketch so that is your answer.
Please note that this would have been different if they had told you your answer should be in Q III. In that case, the answer your calculator gave would have needed to be tweaked. You would have had to project an equivalent cosine in Q III which would have meant you would have had to use a -153° meaning θ would have been 360 - 153 or 207°. Notice if you put cos(153°) in your calculator you get -0.8910 and if you put cos(207°) in, you also get -0.8910. So, don't necessarily believe your calculator without drawing a sketch to determine where you are on the unit circle.
