If \(0 < \theta < \frac{\pi}{2}\) and \(\sqrt{3} \cos \theta - \sin \theta = \frac{1}{3},\) then find \(\sqrt{3} \sin \theta + \cos \theta.\)
+128407 √3 cos θ - sin θ = 1/3
√3 cos θ = sin θ +1/3 square both sides
3cos^2θ = sin^2θ + (2/3)sin θ + 1/9
3 (1 - sin^2θ) = sin^2 θ + (2/3)sin θ + 1/9
3 - 3sin^2 θ = sin^2θ + (2/3)sin θ + 1/9 rearrange as
4sin^2 θ + (2/3) sinθ - 26/9 = 0 multiply through by 9
36sin^2θ +6sin θ - 26 = 0
18 sin^2 θ + 3sin θ - 13 = 0 let sin θ = x
18x^2 + 3x - 13 = 0
Using the quadratic formula and taking the poisitve result [ since the sine is positive here, we get that ]
x = sin θ = - 3 + √[ 3^2 + 4* 18 * 13]
______________________ ≈ .77
2 * 18
Using the sine inverse we get that
arcsin (.77) = θ ≈ .88 rads
So
√3 sin (.88) + cos (.88) ≈ 1.972
+2862 I do not understand any of those trig gibberish yet, but nice job Cphill! One day I will learn the dark secrets of sine cosine and tangent!
