Answer:
x = n(π/15) +{arcsin(-0.348612)/30, π/30 -arcsin(-0.348612)/30}
for n in Integers
Explanation:
You want to find x that satisfies the equation ...
5·sin(30x) -cos(60x) = -2.5
Substitution
The equation is easier to see as a quadratic if we let z = sin(30x). Using the trig identity cos(2x) = 1 -2sin(x)², the equation becomes ...
5z -(1 -2z²) = -2.5
4z² +10z +3 = 0
Quadratic
Using the quadratic formula we find solutions for z to be ...
z = (-10 ±√(10² -4(4)(3)))/(2·4) = (-10 ±√52)/8
Only z values with a magnitude less than 1 are suitable, so ...
z = (-5+√13)/4 ≈ -0.348612
Angles
Then the values of x are ...
sin(30x) = -0.348612
x = (arcsin(-0.348612) +2πn)/30 or ((π -arcsin(-0.348612)) +2πn)/30
x = n(π/15) +{arcsin(-0.348612)/30, π/30 -arcsin(-0.348612)/30}
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Additional comment
A few numerical values are shown in the attachments. They include ...
{0.1166, 0.1976, 0.3260, 0.4070, 0.5355, 0.6164, 0.7449, 0.8259, ...}
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