cos(56°-3x) = -0.79 in the range 0 < x < 360°
We use the inverse cosine on a calculator with |-0.79| or +0.79
to find the reference angle.
(Caution: don't use the given negative value, as it will not give
you the reference angle, which you need to get values between 0°
and 360°. When you study principle values of inverse trig functions
you will understand why you should not use negative values.)
The reference angle for 56°-3x is 37.81448851°
There are two angles between 0° and 360° which have this reference angle.
One is in the 2nd quadrant and one is in the 3rd quadrant, since those
are the two quadrants in which the cosine is negative.
To get the angle in the 2nd quadrant, subtract 37.81448851° from 180°,
getting 142.1855115°. We can add any positive or negative or zero
multiple of 360° to this value and we will still have an angle whose
cosine is -0.79. So all such angles are 37.81448851°+360°k, where
k is any integer.
To get the angle in the 3rd quadrant, add 37.81448851° to 180°,
getting 217.8144885°. Also we can add any positive or negative or zero
multiple of 360° to this value and we will still have an angle whose
cosine is -0.79. So all such angles are 217.8144885°+360°k, where
k is any integer.
So we have two equations:
56°-3x = 142.1855115°+360°k and 56°-3x = 217.8144885°+360°k
-3x = 86.18551149°+360°k -3x = 161.8144885°+360°k
x = -28.72850383°-120°k x = -53.93816284-120°k
To get the answers between 0° and 360°, we have these inequalities:
0° < -28.72850383°-120°k < 360° 0° < -53.93816284-120°k < 360°
Taking the first one. We first
solve this inequality:
28.72850383° < -120°k < 388.7285038°
-.2394041986 > k > -3.239404198
So we take k = -3,-2, and -1 as the only
integers between those decimal values:
Therefore
using k=-3
x = -28.72850383°-120°(-3)
x = 331.2714962°
using k=-2
x = -28.72850383°-120°(-2)
x = 211.2714962°
using k=-1
x = -28.72850383°-120°(-1)
x = 91.27149617°
Now we take the other equation:
x = -53.93816284-120°k
First solve the inequality for k:
0° < -53.93816284-120°k < 360°
53.93816284° < -120°k < 413.9381628°
-.4494846903 > k > -3.44948469
So we take k = -3,-2, and -1 as the only
integers between those decimal values:
x = -53.93816284°-120°k
Using k=-3,
x = -53.93816284°-120°(-3)
x = 306.0618372°
Using k=-2,
x = -53.93816284°-120°(-2)
x = 186.0618372°
Using k=-1,
x = -53.93816284°-120°(-1)
x = 66.0618372°
So there are six solutions between 0° and 360°:
66.06183716°
91.27149617°
186.0618372°
211.2714962°
306.0618372°
331.2714962°
Edwin
