sin(x)sec(x)-2sin(x)=0
solved over {0,3pi}
sin(x)sec(x)-2sin(x)=0
sinx(secx - 2) = 0
sinx = 0 or secx = 2
if sinx = 0 , x = 0, π, 2π, 3π for your given domain
if secx = 2, then
cosx = 1/2
x = π/3, 5π/3, 7π/3
To solve the equation sin(x)sec(x)-2sin(x) = 0 over the interval {0, 3π}, we can follow these steps:
Step 1: Rewrite the equation using trigonometric identities.
sin(x)sec(x) - 2sin(x) = 0
Step 2: Apply the reciprocal identity for sec(x).
sin(x)(1/cos(x)) - 2sin(x) = 0
Step 3: Multiply both sides of the equation by cos(x) to get rid of the denominator.
sin(x) - 2sin(x)cos(x) = 0
Step 4: Factor out sin(x) from the equation.
sin(x)(1 - 2cos(x)) = 0
Step 5: Set each factor equal to zero and solve separately.
For sin(x) = 0:
x = 0, π, 2π
For 1 - 2cos(x) = 0:
2cos(x) = 1
cos(x) = 1/2
To find the values of x over the interval {0, 3π}, we need to consider the domain of sin(x) and cos(x).
For sin(x) = 0, the solutions x = 0, π, 2π are within the interval {0, 3π}.
For cos(x) = 1/2, we can use the unit circle or inverse cosine to find the solutions within the interval {0, 3π}.
cos(x) = 1/2 at x = π/3, 5π/3
Therefore, the solutions to the equation sin(x)sec(x) - 2sin(x) = 0 over the interval {0, 3π} are x = 0, π, 2π, π/3, and 5π/3.
To solve the equation sin(x)sec(x) - 2sin(x) = 0 over the interval {0, 3π}, follow these steps:
Step 1: Simplify the equation
Start by factoring out sin(x) from the equation:
sin(x)(sec(x) - 2) = 0
Step 2: Set each factor equal to zero
Since we know that a product is equal to zero if one or more of its factors is zero, we can set each factor equal to zero and solve for x separately.
Set sin(x) = 0:
sin(x) = 0
This equation is true when x = 0, π, and 2π.
Set sec(x) - 2 = 0:
sec(x) - 2 = 0
sec(x) = 2
Step 3: Solve sec(x) = 2
Recall that sec(x) is the reciprocal of cosine(x), so we can rewrite the equation as:
1/cos(x) = 2
To solve for x, we need to find the angle whose cosine is equal to 1/2.
cos(x) = 1/2
Using the unit circle or trigonometric identities, we find that the solutions to this equation are x = π/3 and x = 5π/3.
Step 4: Combine all the solutions
The solutions to the equation sin(x)sec(x) - 2sin(x) = 0 over the interval {0, 3π} are:
x = 0, π, 2π, π/3, and 5π/3.