Find all values of x in the interval [0, 2π] that satisfy the equation. (Enter your answers as a comma-separated list.)
3 sin(2x) = 3 cos(x)
Given an angle of 30 degrees, find the minimum initial speed of a cannon ball that travels in a horizontal distance of 15000 mi.
sin 2x = 2 sin x cos x
so
6 sin x cos x = 3 cos x
right off, if cos x = 0, we have solutions. That is when
x = pi/2 and x = 3 pi/2
also when sin x=1/2
that is x = pi/6 and x = 5 pi/6
Kadudu, see where you posted below
To find the values of x that satisfy the equation 3sin(2x) = 3cos(x) in the interval [0, 2π], we can use the trigonometric identities and solve for x.
Step 1: Divide both sides of the equation by 3:
sin(2x) = cos(x)
Step 2: Use the double-angle identity for sine:
sin(2x) = 1 - 2sin^2(x)
Step 3: Substitute the equation in Step 2 into the equation in Step 1:
1 - 2sin^2(x) = cos(x)
Step 4: Rearrange the equation:
2sin^2(x) + cos(x) - 1 = 0
Step 5: Notice that the equation in Step 4 is a quadratic equation in terms of sin(x). Set sin(x) as a variable and solve the quadratic equation using factoring, quadratic formula, or any suitable method.
By factoring, we can rewrite the equation as:
(2sin(x) - 1)(sin(x) + 1) = 0
Setting each factor equal to zero, we get two possible solutions:
2sin(x) - 1 = 0 --> sin(x) = 1/2
sin(x) + 1 = 0 --> sin(x) = -1
Step 6: Solve each equation separately to find the values of x.
a) For sin(x) = 1/2:
In the interval [0, 2π], the values of x that satisfy sin(x) = 1/2 are π/6 and 5π/6.
b) For sin(x) = -1:
In the interval [0, 2π], the value of x that satisfies sin(x) = -1 is 3π/2.
Step 7: Combine the solutions from Step 6. Since the question asks for all the values of x that satisfy the equation, we include all the values found above. Therefore, the values of x in the interval [0, 2π] that satisfy the equation 3sin(2x) = 3cos(x) are π/6, 5π/6, and 3π/2.
So the solution is: x = π/6, 5π/6, 3π/2.