+129896 We might also use a trig identity to solve this....note that sin(pi/6)*cosx-cos(pi/6)*sinx= sin(pi/6 -x)......so we have
sin(pi/6 - x) = -1/2
Since the sinθ = -1/2 when θ = 7/6pi, we can solve (pi/6 - x) = 7/6pi, and find that x = -pi. But, -pi is the same coterminal angle as pi. So x = pi is one answer (as Aziz found). As a check, note that sin(pi/6 - pi) = sin (-5/6pi) = -1/2
Also, sinθ = -1/2 when θ = 11/6pi. And in a like manner, we have (pi/6 - x) = 11/6pi, and we find that x = -10/6pi or -5/3pi. But -5/3pi is the same coterminal angle as pi/3... (again, as Aziz found). As a check, note that sin(pi/6 - pi/3) = -1/2.
Using the identity saves a bit of work....but you also obtain answers that you must "convert" to stay within the specified interval....!!! Your choice....
+4473 sin(pi/6)*cosx - cos(pi/6)*sinx = -1/2; interval [0, 2pi]
Using the unit circle, we know that sin(pi/6) = 1/2 & cos(pi/6) = sqrt(3)/2 -->
1/2*cosx - sqrt(3)/2*sinx = -1/2
If x = pi/2 --> 1/2*cos(pi/2) - sqrt(3)/2*sin(pi/2) = 0 - sqrt(3)/2*1 = 0 - sqrt(3)/2 = -sqrt(3)/2
If x = pi/3 --> 1/2*cos(pi/3) - sqrt(3)/2*sin(pi/3) = 1/2*(1/2) - sqrt(3)/2*sqrt(3)/2 = 1/4 - 3/4 = -2/4 = -1/2
If x = pi/4 --> 1/2*cos(pi/4) - sqrt(3)/2*sin(pi/4) = 1/2*(sqrt(2)/2) - sqrt(3)/2*(sqrt(2)/2) = sqrt2/4 - sqrt6/4 = [sqrt(2) - sqrt(6)] / 4 = -0.259
If x = pi --> 1/2*cos(pi) - sqrt(3)/2*sin(pi) = 1/2*(-1) - sqrt(3)/2*0 = -1/2
If x = 2pi --> 1/2*cos(2pi) - sqrt(3)/2*sin(2pi) = 1/2*(1) - sqrt(3)/2*0 = 1/2
Note: We have x = pi/3 & x = pi so far, but we can also consider 2pi/3, 4pi/3, etc.
If x = 2pi/3, we notice that we would get -1/4 - sqrt(3)/2*sqrt(3)/2 = -1/4 - 3/4 = -4/4 = 1
If x = 4pi/3, -1/4 - sqrt(3)/2*-sqrt(3)/2 = -1/4 + 3/4 = 2/4 = 1/2
If x = 5pi/3, 1/4 - sqrt(3)/2*-sqrt(3)/2 = 1/4 + 3/4 = 4/4 = 1.
So, x = pi/3 & x = pi satisfy the equation.
+129896 We might also use a trig identity to solve this....note that sin(pi/6)*cosx-cos(pi/6)*sinx= sin(pi/6 -x)......so we have
sin(pi/6 - x) = -1/2
Since the sinθ = -1/2 when θ = 7/6pi, we can solve (pi/6 - x) = 7/6pi, and find that x = -pi. But, -pi is the same coterminal angle as pi. So x = pi is one answer (as Aziz found). As a check, note that sin(pi/6 - pi) = sin (-5/6pi) = -1/2
Also, sinθ = -1/2 when θ = 11/6pi. And in a like manner, we have (pi/6 - x) = 11/6pi, and we find that x = -10/6pi or -5/3pi. But -5/3pi is the same coterminal angle as pi/3... (again, as Aziz found). As a check, note that sin(pi/6 - pi/3) = -1/2.
Using the identity saves a bit of work....but you also obtain answers that you must "convert" to stay within the specified interval....!!! Your choice....
