Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. Round your answers to three decimal places where appropriate.)
sin 2θ + sin θ = 0
sin2x = 2sinx cosx
so, you have
2sinx cosx + sinx = 0
sinx (2cosx+1) = 0
you should be able to sort that out easily, looking for values where
sinx=0 or cosx = -1/2
If a projectile is fired with velocity v0 at an angle θ, then its range, the horizontal distance it travels (in feet), is modeled by the function
R(θ) =
v02 sin 2θ
32
.
If
v0 = 2600 ft/s,
what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft away? (Enter your answers as a comma-separated list. Round your answers to three decimal places.)
Thanks.
To solve the equation sin 2θ + sin θ = 0 using a double or half-angle formula, we can rewrite sin 2θ as 2 sin θ cos θ.
The equation can now be written as:
2 sin θ cos θ + sin θ = 0
Combining like terms, we have:
3 sin θ cos θ = 0
Now we have two possibilities for this equation:
1) sin θ = 0: In this case, θ can be any multiple of π, as sin θ is zero at these points.
2) cos θ = 0: In this case, θ can be any multiple of π/2, as cos θ is zero at these points.
Putting all the values of θ together in the interval [0, 2π), we have the following solutions:
θ = 0, π/2, π, 3π/2, 2π
Therefore, the solutions to the equation sin 2θ + sin θ = 0 in the interval [0, 2π) are:
θ = 0, π/2, π, 3π/2, 2π
To solve the equation sin 2θ + sin θ = 0 using a Double- or Half-Angle Formula in the interval [0, 2π), we can rewrite the equation using the Double-Angle Formula for sine:
sin 2θ + sin θ = 2sin θ cos θ + sin θ = 0
Now, let's factor out sin θ:
sin θ(2cos θ + 1) = 0
Since the product of two factors is zero, one or both of the factors must be zero. Therefore, we have two possible cases:
1. sin θ = 0
This occurs when θ = 0 or when θ = π.
2. 2cos θ + 1 = 0
Subtracting 1 from both sides, we get:
2cos θ = -1
Dividing by 2, we have:
cos θ = -1/2
The solution for cos θ = -1/2 is θ = 2π/3 or θ = 4π/3 in the interval [0, 2π).
Therefore, the solutions for the equation sin 2θ + sin θ = 0 in the interval [0, 2π) are:
θ = 0, θ = π, θ = 2π/3, and θ = 4π/3.