suppose theta is in the interval (90,180 degrees) find the sign of
csc(-theta)
is it positive or negative
= positive???
-theta is in quadrant 3, lower left
In that quadrant the sin is negative so the csc is negative.
for example if theta = 150 deg
then -theta = -150
and csc(-150) = 1/-.5 = -2
To determine the sign of csc(-theta) when theta is in the interval (90, 180 degrees), we need to consider the values of the csc function in that interval.
The csc function is defined as the reciprocal of the sin function. In the interval (90, 180 degrees), the sin function is positive since it corresponds to the values of the sine function in the second quadrant of the unit circle. Therefore, the reciprocal, csc(-theta), will also be positive.
So, the sign of csc(-theta) is positive.
To determine the sign of csc(-θ), where θ is in the interval (90, 180 degrees), we can use the periodicity and symmetry properties of the trigonometric functions.
First, let's understand what csc(-θ) means. The cosecant function (csc) is the reciprocal of the sine function. So, csc(-θ) is equal to 1/sin(-θ).
Since θ is in the interval (90, 180 degrees), we know that it lies in the second quadrant of the unit circle. In the second quadrant, the sine function is positive.
Now, let's consider -θ. When we take the negative of θ, we reflect it across the origin on the Cartesian plane, which means the angle -θ will be in the fourth quadrant of the unit circle.
In the fourth quadrant, the sine function is also positive. Therefore, sin(-θ) is positive.
Since csc(-θ) = 1/sin(-θ), and sin(-θ) is positive, the reciprocal 1/sin(-θ) will also be positive.
Therefore, the sign of csc(-θ) is positive.
In summary, csc(-θ) is positive for θ in the interval (90, 180 degrees).