solve the equation 5sin(θ -pi/6)=8cosθ for values of 0<=θ <=2pi
To solve the equation 5sin(θ - π/6) = 8cosθ for values of 0 ≤ θ ≤ 2π, we can use trigonometric identities and algebraic manipulation. Here's how you can solve it step by step:
1. Start with the given equation: 5sin(θ - π/6) = 8cosθ.
2. Use the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b) to rewrite the left side of the equation:
5(sin(θ)cos(π/6) - cos(θ)sin(π/6)) = 8cosθ.
3. Simplify the expression:
5(cos(π/6)sin(θ) - sin(π/6)cos(θ)) = 8cosθ.
4. Now, use the identity sin(π/6) = 1/2 and cos(π/6) = √3/2:
5(√3/2 * sin(θ) - 1/2 * cos(θ)) = 8cosθ.
5. Distribute the 5 on the left side of the equation:
5√3/2 * sin(θ) - 5/2 * cos(θ) = 8cosθ.
6. Move all terms involving sinθ to one side and terms involving cosθ to the other side:
8cosθ + 5/2 * cos(θ) = 5√3/2 * sin(θ).
7. Factor out cosθ on the left side:
(8 + 5/2) * cos(θ) = 5√3/2 * sin(θ).
8. Simplify the coefficients on both sides:
21/2 * cos(θ) = 5√3/2 * sin(θ).
9. Divide both sides of the equation by 21/2 * sin(θ):
(21/2 * cos(θ)) / (5√3/2 * sin(θ)) = 1.
10. Simplify the expression on both sides by canceling out common factors:
(21/2)/(5√3/2) = 1.
11. Evaluate the expression:
(21/2) / (5√3/2) = (21/2) * (2/(5√3)) = 21 / (5√3).
12. Rationalize the denominator by multiplying both the numerator and the denominator by √3:
(21 / (5√3)) * (√3/√3) = 21√3 / 15 = (7√3) / 5.
So, the solution to the equation 5sin(θ - π/6) = 8cosθ for values of 0 ≤ θ ≤ 2π is θ = arcsin((7√3) / 5). Evaluating this angle in the given range, we can find the specific values.