1. Let \(\theta\) be an angle. Then there exists constants a and b such that \(\cos\left(\theta + 60^{\circ}\right) = a \sin (\theta) +b \cos (\theta)\) for all \(\theta\). Enter a, b in that order.
2. Let \(\theta\) be an angle. Then there exist constants a and b such that \(\sin\left(\theta + \arctan\left(\frac{5}{12} \right)\right) = a \sin (\theta) +b \cos (\theta)\) for all \(\theta\). Enter a, b in that order.
3. Find the greatest possible value of \(4 \sin (\theta) + 3 \cos (\theta)\).
Thanks for the help!
+128475 We are using the angle sum formulas here :
1) cos ( θ + 60°) = cos (60°) sin (θ) - sin(60°)cos(θ) = (1/2)sin(θ) - (√3/2)cos(θ)
So a = 1/2 and b = -√3/2
2) sin ( θ + arctan (5/12) ) = cos (arctan (5/12)) * sin(θ ) + sin (arctan (5/12)) * cos (θ)
Note
cos ( arctan (5/12) ) = 12/sqrt (5^2 + 12^2) = 12/sqrt(169) = 12/13 = a
sin (arctan (5/12) ) = 5 / sqrt (5^2 + 12^2) = 5/sqrt (169) = 5/13 = b
+128475 3) 4sin (θ) + 3cos(θ)
Take the derivative and set to 0
4cos (θ) - 3sin(θ) = 0
4cos (θ) = 3 sin (θ)
(4/3) = sin (θ)/cos(θ)
(4/3) = tan (θ)
arctan (4/3) = θ
So
4 sin ( arctan (4/3)) + 3cos (arctan (4/3) =
4 (4 / sqrt (4^2 + 3^2)) + 3 ( 3/sqrt (4^2 + 3^2) ) =
16 / sqrt (25) + 9 /sqrt (25) =
16/5 + 9/5 =
25/5 =
5
