sin2x-cosx (0<=x<=180)
To find the values of sin(2x) - cos(x) for x between 0 and 180 degrees, we can use the trigonometric identities and properties.
First, let's break down the expression sin(2x).
Using the double angle formula, sin(2x) can be expressed as:
sin(2x) = 2sin(x)cos(x)
Now, let's substitute this into the given expression:
sin(2x) - cos(x) = 2sin(x)cos(x) - cos(x)
Now, we can factor out cos(x) to simplify the expression:
sin(2x) - cos(x) = cos(x)(2sin(x) - 1)
We can now analyze the expression 2sin(x) - 1.
To find out when this expression is zero or positive, we set it equal to zero:
2sin(x) - 1 = 0
Adding 1 to both sides:
2sin(x) = 1
Dividing both sides by 2:
sin(x) = 1/2
Now, we need to find the values of x that satisfy sin(x) = 1/2, within the given range of 0 to 180 degrees.
To find these solutions, we recall that sin(x) = 1/2 for angles x = 30 degrees and x = 150 degrees in the first and second quadrants respectively.
Therefore, for x = 30 degrees, we substitute it back into our original expression:
sin(2x) - cos(x) = sin(60) - cos(30)
Using the values of sin(60) = sqrt(3)/2 and cos(30) = sqrt(3)/2, we get:
sqrt(3)/2 - sqrt(3)/2 = 0
Similarly, for x = 150 degrees, we substitute it back into the original expression:
sin(2x) - cos(x) = sin(300) - cos(150)
Using the values of sin(300) = -sqrt(3)/2 and cos(150) = -sqrt(3)/2, we get:
-sqrt(3)/2 - (-sqrt(3)/2) = 0
So, for x = 30 degrees and x = 150 degrees, the value of sin(2x) - cos(x) is equal to 0.
In summary, the values of sin(2x) - cos(x) for x between 0 and 180 degrees are 0 when x = 30 degrees and x = 150 degrees.