Question

Find the radian measure of all angles whose cosine is -0.89

(One of the answers is 3.62, but shouldn’t we add pi instead of minus 2pi) (I have a test tomorrow please answer if you know!)

Answer 1

Check the picture below.

so going on the 1st revolution from 0 to 2π, we get those two angles you see in the picture, the cosine is positive on I and IV Quadrants, whilst negative on the II and III Quadrants, so any angles with that negative cosine will be only on those two Quadrants.

when getting an angle or a trig function, make sure your calculator is in the correct mode, because plugging in cos(-0.89) in Degree mode, it's going to assume you mean to find the cosine of -0.89 Degrees, and same will happen the other way around, so make sure your mode is correct, so the input is understood correctly.

the cos⁻¹ function has a range of 0 to π, so any angles it's going to spit out will be on those Quadrants, we get the one in the III Quadrant in the picture by simply using the reference angle from the II Quadrant, the reference angle is about 0.47359 radians, so if we add that to π, we get that many radians.

now, if we want to include All revolutions around the circle, we can say

`\approx ~~ \begin{cases} 2.668+2\pi n\\\\ 3.615+2\pi n \end{cases}\qquad n\in \mathbb{Z};n\geqslant 0`