Suppose that 0degrees<theta<180degrees. what is the sign of each of the following?
a) cos theta/2
b)sin(-theta)
cos theta/2 is positive and sin(-theta) is negative
How do you know by looking at them?
Sketch a graph
If theta is between 0 and 180 then theta/2 is between 0 and 90. Therefore in the first quadrant.
In the first quadrant, all trig functions are positive.
If theta is between 0 and 180 then negative theta is between 180 and 360 in the lower two quadrants, quadrants 3 and 4. In those quadrants y is negative so sin theta is negative.
oh okay! I get it now, thanks!:)
To determine the signs of each trigonometric function, we need to understand the quadrants in which the angle lies.
a) cos(theta/2):
To find the sign of cos(theta/2), we first need to determine the quadrant in which the angle (theta/2) lies.
Since 0 degrees < theta < 180 degrees, the angle (theta/2) will be in the first or second quadrant.
In the first quadrant (0 to 90 degrees), both cosine and sine are positive. Therefore, cos(theta/2) will be positive.
In the second quadrant (90 to 180 degrees), the cosine is negative, while the sine is positive. However, since theta/2 is less than 90 degrees, cos(theta/2) will still be positive in this range.
So, in both cases, cos(theta/2) will be positive.
b) sin(-theta):
To find the sign of sin(-theta), we need to consider the negative angle (-theta).
Since 0 degrees < theta < 180 degrees, the negative angle (-theta) will lie in the third or fourth quadrant.
In the third quadrant (180 to 270 degrees), both sine and cosine are negative. Therefore, sin(-theta) will be negative.
In the fourth quadrant (270 to 360 degrees), the sine is positive, while the cosine is negative. However, since -theta is greater than 180 degrees, sin(-theta) will still be negative in this range.
So, in both cases, sin(-theta) will be negative.