Question

Find all possible solutions in the range 0°≤ x ≤ 180° of the following equations:

(a) cos x = sqrt(2) / 2

(b) tan x = sqrt(3) / 3

(c) sin x = sin60°

(d) cos x = -1/2

(e) sin x = -1/2

Answer 1

All of the trig ratios shown are based on "standard" angles. We used to have math kits that contained a compass, a protractor and two triangles.
One had angles 45-45-90°, the other had 30-60-90°
The ratio of sides of the 30-60-90 were 1 : √3 : 2
and for the 45-45-90 it is 1 : 1 : √2

I strongly suggest you make a sketch of these triangles in your notebook and label the sides.
Now you can easily find and recognize all six trig ratios of angles 30, 45, and 60° angles.
You should also know the CAST rule, which tells you were trig ratios of certain angles are + or - .

e.g. d)
cos x = -1/2
from your triangles, you can see that cos 60° = +1/2
but cosx is negative, so the angle could be in quadrants II or III using the CAST rule.
in II, x = 180-60 = 120°
in III, x = 180+60 = 240° , but you wanted to have ≤ 180°
so x = 120°

All can be checked with a calculator

c) is different but the easiest of the bunch.
sinx = sin 60
Looks like we started with x = 60 , and somebody took the sine of both sides.
Remember the old grade 9 rule: "whatever you do to one side of an equation, you must do to the other side".

for a) cosx = √2/2, or 1/√2
look at your 45-45-90 triangle, isn't
cos 45° = 1/√2 ???
so x = 45°

Let me know what you get for the others.