trigonometry help?

Question

Express 2sin t + [2*3^ (0.5)] cos t in the form R sin (t + alpha), where 0< alpha< 1/2 of pi

^ means to the power of .

is my answer correct?

(root of 16) sin (t+ 0.25 of pi)


next question is; wat is the least value of 1 / [ 10+2sint+(2*3^0.5) cos t ]

as t varies?

Thanks a lot in advance

Answer 1

2 sin t + 2√3 cos t = R sin(t + α)
2 sin t + 2√3 cos t = R sin t cos α + R sin α cos t

setting coefficients equal,
R cos α = 2
R sin α = 2√3
R²cos² α + R²sin² α = 2² + (2√3)²
R²(cos² α + sin² α) = 4 + 12
R² = 16
R = ±4

if cos α = 1/2 and sin α = √3 / 2, then α = π/3, not π/4.

so 2 sin t + 2√3 cos t = 4 sin(t + π/3), which has a max value of 4 when t = π/6, so 10 + that has max of 14, which makes least value of the fraction 1/14.