Question 1156790
the quadrant that the angle is in is based on the sign of the trig function.
in the first quadrant, all trig functions are positive.
you concentrate on sine, cosine, and tangent.
since cosecant is 1/sine and secant is 1/cosine and cotangent is 1/tangent, they will follow the sign of their reciprocals of sine, cosine, and tangent.
the hypotenuse of the triangle formed by the angle is always positive because it is the square root of (x^2 + y^2) which is always positive because x^2 and y^2 are always positive, regardless of the underlying sign of x and y.
the rules are as follows:
sine, cosine, and tangent are all positive in the first quadrant.
in the second quadrant, sine is positive and cosine is negative and tangent is negative because it is equal to sine / cosine which is a positive divided by a negative, therefore negative.
in the third quadrant, sine is negative and cosine is negative and tangent is positive because tangent is sine / cosine which is a negative divided by a negative which is positive.
in the fourth quadrant, sine is negative and cosine is positive and tangent is negative because it is equal to sine / cosine which is a negative divided by a positive which is negative.
here's a reference on the unit circle that you might find informative.
<a href = "https://www.purplemath.com/modules/unitcirc.htm" target = "_blank">https://www.purplemath.com/modules/unitcirc.htm</a>
if you have any further questions regarding how to determine the quadrant that the angle is in, feel free to email me at dtheophilis@gmail.com.
keep in mind that, in the first quadrant, all trig functions are positive and sine and cosine are always between 0 and 1, while tangent can be any value depending on the ratio of sine / cosine.
the quadrant that the angle is in is determined by the sign of the trig function that the angle is derived from.
the rules for what quadrant the angle is in are as follows.
if the sine is positive, the angle is in the first or second quadrant.
if the sine is negative, the angle is in the third or fourth quadrant.
if the cosine is positive, the angle is in the first or fourth quadrant.
if the cosine is negative, the angle is in the second or third quadrant.
if the tangent is positive, the angle is in the first or third quadrant.
if the tangent is negative, the angle is in the second or fourth quadrant.
here's another reference that might be helpful.
<a href = "https://www.purplemath.com/modules/quadangs2.htm" target = "_blank">https://www.purplemath.com/modules/quadangs2.htm</a>