SOLUTION: How do you figure out which quadrant the terminal side of each angle lies in based on if sin/cos/tan, etc. is greater than or less than theta? For example, Sec(theta)=13/12, sin

Question 1156790: How do you figure out which quadrant the terminal side of each angle lies in based on if sin/cos/tan, etc. is greater than or less than theta?
For example, Sec(theta)=13/12, sin(theta)<0
Found 2 solutions by Theo, Edwin McCravy:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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.
https://www.purplemath.com/modules/unitcirc.htm
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.
https://www.purplemath.com/modules/quadangs2.htm
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

Learn the sentence: "All Students Take Calculus" to remember "All S T C" For the quadrants, think of this as though it were an x,y coordinate system: S|A T|C The major three trig functions are Sine, Cosine, and Tangent, the others are their reciprocals. This memory device only works with the major three trig functions, Sine, Cosine and Tangent. "All" reminds us that ALL trig functions are POSITIVE in the first quadrant QI. "STUDENTS" reminds us that of the major three trig functions, the SINE is POSITIVE in the second quadrant QII, and the other two major trig functions are NEGATIVE there. "TAKE" reminds us that of the major three trig functions, the TANGENT is POSITIVE in the third quadrant QIII, and the other two major trig functions are NEGATIVE there. "CALCULUS" reminds us that of the major three trig functions, the COSINE is POSITIVE in the fourth quadrant QIV, and the other two major trig functions are NEGATIVE there. If the major trig function, the SINE, is positive or negative in a quadrant, its reciprocal, the COSECANT is also positive or negative there. If the major trig function, the COSINE, is positive or negative in a quadrant, its reciprocal, the SECANT is also positive or negative there. If the major trig function, the TANGENT, is positive or negative in a quadrant, its reciprocal, the COTANGENT is also positive or negative there. Now if we have learned that, let's go through what this information tells us: Sec(theta)=13/12, sin(theta)<0 13/12 is a POSITIVE number, and the SECANT is not one of the major trig functions. However we know that SECANT is the reciprocal of the major trig function, the COSINE and "COSINE" begins with "C" and so does "Calculus". That tells us that theta is either in QI or QIV because those are the only two quadrants that the COSINE is POSITIVE in. The next one, "sin(theta)<0", tells us that the SINE is NEGATIVE, and the SINE is NEGATIVE in QII and QIV. So now we know that theta is in QIV. So we draw an angle θ with its terminal side in QIV, and indicate its counter- clockwise rotation from the right side of the x-axis around to its terminal side in QIV by a red arrow: Draw a vertical line from the end of the terminal side to the x-axis: Now we know that the SECANT is HYPOTENUSE/ADJACENT because it's the reciprocal of the COSINE which in ADJACENT/HYPOTENUSE, so we put the numerator of 13/12, which is 13 on the HYPOTENUSE, which is denoted by the letter "r" and we put the denominator of 13/12, which is 12, on the ADJACENT side, which is denoted by the letter "x". The horizontal value of y will need to be calculated. Next we need to calculate the OPPOSITE side or the y-value. We use the Pythagorean theorem: We take the negative sign because y goes down from the x-axis. So y=-5. Now we can find any of the trig functions: Edwin

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