Solve sin(2è) < cos(è). Give an exact answer.
To solve the inequality sin(2è) < cos(è), we can use the fact that sin(2è) = 2sin(è)cos(è).
So, we have 2sin(è)cos(è) < cos(è).
We can now consider two cases:
Case 1: cos(è) ≠ 0
In this case, we can divide both sides of the inequality by cos(è) (since it is a positive quantity) to get:
2sin(è) < 1.
Now, we can divide both sides by 2 to isolate sin(è):
sin(è) < 1/2.
To find the exact values of è in this case, we can refer to the unit circle. Recall that sin(è) is positive in the first and second quadrants. In the first quadrant, sin(è) < 1/2 for angles less than 30 degrees or π/6 radians. In the second quadrant, sin(è) < 1/2 for angles greater than 150 degrees or 5π/6 radians.
Therefore, in this case, the exact solution is:
0 ≤ è < 30 degrees or 0 ≤ è < π/6 radians,
or
150 degrees < è < 360 degrees or 5π/6 < è < 2π.
Case 2: cos(è) = 0
If cos(è) = 0, we cannot divide by zero, so we need to consider this as a separate case.
When cos(è) = 0, we have two possibilities:
For è = π/2 radians or 90 degrees, sin(2è) = sin(π) = 0, which is equal to cos(è). So, this value satisfies the inequality.
Similarly, for è = 3π/2 radians or 270 degrees, sin(2è) = sin(3π) = 0, which is equal to cos(è). So, this value also satisfies the inequality.
Therefore, in this case, the exact solutions are:
è = π/2 radians or 90 degrees, and
è = 3π/2 radians or 270 degrees.