Solve the equation in the interval [0, 2pi].
2sin(t)cos(t)-cos(t)+2sin(t)-1=0
Answer needs to be formatted as x={Insert answers here}pi
If the answer is something like pi/6 it simply goes in as 1/6th since pi is already given at the end.
2sin(t)cos(t)-cos(t)+2sin(t)-1=0
cost(2sint - 1) + (2sint - 1) = 0
(2sint - 1)(cost + 1) = 0
sint = 1/2 or cost = -1
if sint = 1/2
t = π/6 or t = 5π/6
if cost = -1, look at your cosine curve,
t = π
To solve the given equation 2sin(t)cos(t) - cos(t) + 2sin(t) - 1 = 0 in the interval [0, 2π], we will break it down step by step:
Step 1: Combine like terms:
2sin(t)cos(t) + 2sin(t) - cos(t) - 1 = 0
Step 2: Rearrange the terms:
2sin(t)cos(t) + 2sin(t) - cos(t) = 1
Step 3: Factor out sin(t):
sin(t)(2cos(t) + 2) - cos(t) = 1
Step 4: Combine terms and rearrange:
2sin(t)cos(t) + sin(t) - cos(t) - 1 = 0
Step 5: Factor by grouping:
sin(t)(2cos(t) + 1) - 1(2cos(t) + 1) = 0
Step 6: Apply the zero-product property:
sin(t) = 1 or 2cos(t) + 1 = 0
Step 7: Solve for sin(t):
sin(t) = 1
Using the unit circle or trigonometric identity, we know that sin(t) = 1 when t = π/2.
Step 8: Solve for cos(t):
2cos(t) + 1 = 0
Subtracting 1 from both sides:
2cos(t) = -1
Dividing by 2:
cos(t) = -1/2
Using the unit circle or trigonometric identity, we know that cos(t) = -1/2 when t = 2π/3 or t = 4π/3.
So, the solutions in the interval [0, 2π] are:
t = π/2, 2π/3, and 4π/3.
To format the answers as requested, we convert the angles into fractions of π:
t = π/2, 2π/3 = 2/3π, and 4π/3 = 4/3π.
Therefore, the final answer is:
t = {π/2, 2/3π, 4/3π}