Quarry School

Signs of the six functions in the four quadrants

Explain it like I am five

Picture a building's side view: a hallway crossing an elevator shaft. Steps right of the shaft are positive, steps left negative; floors above the hallway positive, basement floors negative. The hallway and shaft are the x-axis and y-axis; they split the picture into four quadrants, numbered I, II, III, IV counterclockwise from the upper right. A coordinate is a signed position: its size says how far, its sign says which side.

On the unit circle sin t is the y-coordinate and cos t the x-coordinate, so sine carries the up-down sign and cosine the right-left sign.

Example: the point (−0.96, 0.28) is left and up, in Quadrant II. sin t = 0.28, positive. cos t = −0.96, negative. tan t = yx = 0.28−0.96, about −0.29, since positive divided by negative is negative. Each flip, 1 over its partner, keeps the partner's sign: csc t = 1sint, about 3.57, is positive, while sec t = 1cost and cot t = 1tant are negative. Only sine and cosecant are positive: the S of All Students Take Calculus.

In plain words

Think of a town map with the center of town at the origin, the point (0, 0). East is positive x, west is negative x, north is positive y, and south is negative y. The axes divide the map into four regions called quadrants. Their names are I, II, III and IV, starting at the upper right and going counterclockwise. On the unit circle, sine reads your north or south position and cosine reads your east or west position. You can decide their signs without finding their sizes. The other four functions divide or flip those coordinates, so the sign rules for fractions finish the job.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read x from left or right and y from below or above before building the other signs.
Reminder
  • Signs of a quotient. Matching signs give a positive quotient: −4−3 = 43. Different signs give a negative quotient: 4−3 = −43.
  • Reciprocal signs. A reciprocal keeps the sign of a nonzero number: 1 ÷ (−4) = −14.
  • Axis values. At (0, 1), cosine is 0, sine is 1, tangent is undefined and cotangent is 0. A 0 on the bottom, rather than on top, causes undefined.
QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read x from left or right and y from below or above before building the other signs.
input Quadrantoutput sin t = yI+II+III−IV−
Read the output entry in the column under its input or category.
input Quadrantoutput cos t = xI+II−III−IV+
Read the output entry in the column under its input or category.
input Quadrantoutput tan t = [[y|x]]I+II−III+IV−
Read the output entry in the column under its input or category.
input Quadrantoutput cot t = [[x|y]]I+II−III+IV−
Read the output entry in the column under its input or category.
input Quadrantoutput sec t = [[1|x]]I+II−III−IV+
Read the output entry in the column under its input or category.
input Quadrantoutput csc t = [[1|y]]I+II+III−IV−
Read the output entry in the column under its input or category.
Why it works. Quadrant numbering is a convention chosen to follow the direction positive angles turn. Keeping that convention lets everyone read the same sign table. The signs follow from definitions, rather than from the quadrant names: sin t = y and cos t = x. A reciprocal keeps its nonzero partner's sign because 1 is positive. Thus csc follows sin and sec follows cos. A quotient of two nonzero coordinates is positive when their signs match and negative when they differ. Therefore tan = yx and cot = xy are positive in I and III, and negative in II and IV.
RuleInside a quadrant: I has all six positive. II has sin and csc positive. III has tan and cot positive. IV has cos and sec positive. Each unlisted function is negative. Axis points need their actual coordinates, because a value may be 0 or undefined.
The same idea, five ways
Say it

Sine follows y, cosine follows x, their reciprocals keep the signs, and tangent and cotangent compare the two signs.

Write it

The signs of the six trigonometric functions come from the signs of the terminal point's two coordinates.

In math
  • sin t = y
  • cos t = x
  • tan t = yx
  • cot t = xy
  • sec t = 1x
  • csc t = 1y
Like

A town map tells you east or west and north or south before it tells you a distance.

See it
QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read x from left or right and y from below or above before building the other signs.
The same idea, other ways
As a map

Put your finger in the lower left of a town map. Both east or west position x and north or south position y are negative. That immediately gives negative cosine and sine. The matching signs make tangent and cotangent positive.

x = cos θy = sin θx < 0, y < 0
The lower left has two negative coordinates.
As arithmetic

The signs of −4−3 and −3−4 are positive because the two negatives cancel. The signs of 1−3 and 1−4 are negative. The same four calculations explain tangent, cotangent, secant and cosecant in Quadrant III.

−4−3 > 0
−3−4 > 0
1−3 < 0
1−4 < 0
Matching signs give positive quotients; reciprocals retain the negative sign.
As a memory device

Say All Students Take Calculus while moving counterclockwise through I, II, III, IV. All six are positive in I. The next words name the positive pair: sine with cosecant, tangent with cotangent, cosine with secant. Rebuild the signs from x and y if the phrase slips your mind.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read x from left or right and y from below or above before building the other signs.
Working backward

A location question reverses the sign table. If cosine is positive, the point is on the right. If cosecant is negative, sine is negative, so the point is below. Right and below together identify Quadrant IV. Each condition removes locations until only a shared location remains.

x = cos θy = sin θright and below
Positive cosine and negative cosecant put the point in the lower right.
Quadrantsin t = ycos t = xtan t = yxcot t = xysec t = 1xcsc t = 1y
I++++++
II+−−−−+
III−−++−−
IV−+−−+−
.1Sine

Sine uses the terminal point's coordinates. Read its height, like a floor number above or below street level.

  • sin t = y
  • I and II: positive. III and IV: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θsin t = y
A point above the horizontal axis has positive height.
Worked exampleReading the sign of sine

θ is in Quadrant II. What is the sign of sin θ? The question asks for the output's sign.

x = cos θy = sin θsin t = y
A point above the horizontal axis has positive height.
  1. The point is above the x-axis, so y > 0. Since sin θ = y, sine is positive.Sine reads height rather than horizontal position.
Answer
sin θ > 0
Check A point above the horizontal axis has positive height. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use sin t = y and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
.2Cosine

Cosine uses the terminal point's coordinates. Read its left or right position, like east or west of town.

  • cos t = x
  • I and IV: positive. II and III: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θcos t = x
A point right of the vertical axis has positive x.
Worked exampleReading the sign of cosine

θ is in Quadrant IV. What is the sign of cos θ? The question asks for the output's sign.

x = cos θy = sin θcos t = x
A point right of the vertical axis has positive x.
  1. The point is right of the y-axis, so x > 0. Since cos θ = x, cosine is positive.Cosine reads horizontal position.
Answer
cos θ > 0
Check A point right of the vertical axis has positive x. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use cos t = x and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
.3Tangent

Tangent uses the terminal point's coordinates. Compare two directions by dividing. Matching directions give a positive ratio; opposite directions give a negative ratio.

  • tan t = yx
  • I and III: positive. II and IV: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θtan t = [[y|x]]
The ratio of the two negative legs is positive.
Worked exampleReading the sign of tangent

θ is in Quadrant III. What is the sign of tan θ? The question asks for the output's sign.

x = cos θy = sin θtan t = [[y|x]]
The ratio of the two negative legs is positive.
  1. Here x < 0 and y < 0. Their quotient yx is positive.Two matching negative signs give a positive quotient.
Answer
tan θ > 0
Check The ratio of the two negative legs is positive. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use tan t = yx and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
.4Cotangent

Cotangent uses the terminal point's coordinates. Compare two directions by dividing. Matching directions give a positive ratio; opposite directions give a negative ratio.

  • cot t = xy
  • I and III: positive. II and IV: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θcot t = [[x|y]]
The x and y legs have different signs.
Worked exampleReading the sign of cotangent

θ is in Quadrant II. What is the sign of cot θ? The question asks for the output's sign.

x = cos θy = sin θcot t = [[x|y]]
The x and y legs have different signs.
  1. Here x < 0 and y > 0. Their quotient xy is negative.Different signs give a negative quotient.
Answer
cot θ < 0
Check The x and y legs have different signs. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use cot t = xy and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
.5Secant

Secant uses the terminal point's coordinates. Flip one coordinate. A positive coordinate stays positive after flipping and a negative coordinate stays negative.

  • sec t = 1x
  • I and IV: positive. II and III: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θsec t = [[1|x]]
Secant has the sign of the negative x-coordinate.
Worked exampleReading the sign of secant

θ is in Quadrant III. What is the sign of sec θ? The question asks for the output's sign.

x = cos θy = sin θsec t = [[1|x]]
Secant has the sign of the negative x-coordinate.
  1. Here x < 0, so 1x is negative.Positive 1 divided by a negative number is negative.
Answer
sec θ < 0
Check Secant has the sign of the negative x-coordinate. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use sec t = 1x and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
.6Cosecant

Cosecant uses the terminal point's coordinates. Flip one coordinate. A positive coordinate stays positive after flipping and a negative coordinate stays negative.

  • csc t = 1y
  • I and II: positive. III and IV: negative.
  • Inside a quadrant, x and y are both nonzero, so this function is defined.
x = cos θy = sin θcsc t = [[1|y]]
Cosecant has the sign of the negative y-coordinate.
Worked exampleReading the sign of cosecant

θ is in Quadrant IV. What is the sign of csc θ? The question asks for the output's sign.

x = cos θy = sin θcsc t = [[1|y]]
Cosecant has the sign of the negative y-coordinate.
  1. Here y < 0, so 1y is negative.Positive 1 divided by a negative number is negative.
Answer
csc θ < 0
Check Cosecant has the sign of the negative y-coordinate. A point such as the one shown supplies a direct coordinate check.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Decide the sign from the function's name alone.
The name says which coordinates to use; the coordinates' signs determine the answer.
✓ Instead: Use csc t = 1y and check the relevant coordinate signs.
Tips and tricks
  • Point to the relevant coordinate in the picture before writing a sign.
Strategy: step by step
  1. 1. If the question gives a quadrant, locate that region in the picture and write its x and y signs: I (+, +), II (−, +), III (−, −), IV (+, −).
  2. 2. Use sin t = y and cos t = x. A positive sign means an output greater than 0; a negative sign means an output less than 0.
  3. 3. Give csc the sign of sin, and sec the sign of cos. A reciprocal of a nonzero number keeps that number's sign.
  4. 4. Give tan and cot a positive sign when the x and y signs match, and a negative sign when they differ.
  5. 5. If the question gives signs and asks for location, list every quadrant and axis allowed by each condition, then keep their shared locations. For example, cosine positive allows I, IV and the positive x-axis; sine positive allows I, II and the positive y-axis. Do not drop an axis candidate until another condition excludes it.
  6. 6. If a condition says a value is 0 or undefined, check the four axis points before using the quadrant rows. For example, cos t = 0 means x = 0, so the point lies on the y-axis.
  7. 7. Memory device: All Students Take Calculus. In quadrant order, All, Sine, Tangent and Cosine are positive, each named function together with its reciprocal. This cue applies inside quadrants.
  8. Read a reference-table entry in the column under its input or category in the matching picture.
Strategy
Finding signs or recovering a location
1
Is a quadrant given?
YesRead its x and y signs, then use coordinate, quotient and reciprocal definitions.
NoCheck whether an angle or function conditions are given.
↓
2
Are function conditions given?
YesList every quadrant and axis each condition permits, then keep only shared locations.
NoLocate the angle first using quarter-turn boundaries.
↓
3
Does a condition say 0 or undefined?
YesCheck the axis points, including which coordinate is zero.
NoUse positive and negative signs in the quadrant table.
  1. 1. Read whether the given information is a quadrant, an angle or a list of signs.
  2. 2. For a known quadrant, fill in x and y and then build the six signs.
  3. 3. For an angle, locate its terminal side using the next lesson, then use the appropriate quadrant row.
  4. 4. For signs, list all quadrant and axis candidates for each condition and keep only shared locations. If the given conditions force both coordinates nonzero, axes are excluded before using quadrant-only lists.
  5. 5. For a zero or an undefined value, use the axis points and check any remaining condition.
Worked exampleFinding where a terminal point lies from two sign conditions

The terminal point (x, y) of an angle t lies on the unit circle. For each part, find every quadrant or axis direction where the point can lie. (a) csc t > 0 and cot t < 0. (b) sec t < 0 and tan t ≥ 0.

  1. (a) Rewrite the first condition. csc t = 1y, so csc t > 0 means y > 0. That is the same as sin t > 0.csc is the reciprocal of sin, and a reciprocal of a nonzero number keeps that number's sign. csc t is defined only when y ≠ 0.
  2. List the locations where y > 0: quadrant I, quadrant II and the positive y-axis.Sine is positive above the x-axis. The positive y-axis has y = 1 > 0, so it stays a candidate for now.
  3. Rewrite the second condition. cot t = xy < 0. This requires y ≠ 0, and x and y must have different signs. That allows quadrant II (−, +) and quadrant IV (+, −).A quotient is negative when its two signs differ. Cot is undefined on the x-axis, where y = 0.
  4. Test the axis candidate. At the positive y-axis, (0, 1), cot t = 01 = 0. Zero is not negative, so cot t < 0 excludes this point.An axis candidate is dropped only when one of the conditions rules it out. Here the actual coordinates do that.
  5. Keep the shared locations. {I, II, positive y-axis} and {II, IV} have only quadrant II in common.The point must satisfy both conditions at once.
  6. (b) Rewrite sec t = 1x < 0 as x < 0, which is cos t < 0. List the locations: quadrant II, quadrant III and the negative x-axis.sec has the sign of cos. Cosine is negative to the left of the y-axis. The negative x-axis has x = −1, so it is a candidate.
  7. Rewrite tan t = yx ≥ 0. This allows matching signs, which is quadrants I and III. It also allows tan t = 0, which happens when y = 0 and x ≠ 0. That adds the positive x-axis and the negative x-axis.The condition includes 0. Step 6 says to check the axis points whenever a value may be 0. On the y-axis, x = 0, so tan t is undefined there.
  8. Keep the shared locations. {II, III, negative x-axis} and {I, III, positive x-axis, negative x-axis} have quadrant III and the negative x-axis in common.At (−1, 0): sec t = −1 < 0 and tan t = 0 ≥ 0. Both conditions hold, so this axis direction stays in the answer.
Answer
(a) Quadrant II only. (b) Quadrant III or the negative x-axis.
Check (a) Take t = 150°, where (x, y) = (−32, 12). Then csc t = 2 > 0 and cot t = −3 < 0. Both conditions hold. (b) Take t = 225°, where (x, y) = (−22, −22). Then sec t = −2 < 0 and tan t = 1 ≥ 0. Also take t = 180°, where (x, y) = (−1, 0). Then sec t = −1 < 0 and tan t = 0 ≥ 0. All Students Take Calculus agrees inside the quadrants: only sin and csc are positive in II, and only tan and cot are positive in III.

Work to write

  1. csc t > 0 ⇒ y > 0: I, II, positive y-axis
  2. cot t < 0: II, IV (cot = 0 on the positive y-axis, so that axis is excluded)
  3. (a) Quadrant II
  4. sec t < 0 ⇒ x < 0: II, III, negative x-axis
  5. tan t ≥ 0: I, III, positive x-axis, negative x-axis
  6. (b) Quadrant III or the negative x-axis

(a) Quadrant II only. (b) Quadrant III or the negative x-axis.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: signs in a named quadrant

θ is inside Quadrant I. Give all six signs. The question asks which side of 0 each output lies on.

x = cos θy = sin θI
Every expression is built from positive coordinates.
  1. x > 0 and y > 0.Quadrant I is right of the y-axis and above the x-axis.
  2. sin and cos are positive, and their reciprocals csc and sec are positive.Sine reads y, cosine reads x, and reciprocals keep nonzero signs.
  3. tan and cot are positive.Each divides one positive coordinate by the other.
Answer
All six functions are positive.
Check Both coordinates are positive, so no quotient or reciprocal introduces a negative sign.
Rung 2Rung 2: intersect two coordinate signs

An angle inside a quadrant has sin θ < 0 and cos θ < 0. Find its quadrant. The outputs give two location clues.

x = cos θy = sin θx < 0, y < 0
Left and below identify III.
  1. Both sine and cosine are strictly negative, so both coordinates are nonzero. Axis points cannot meet the combined conditions.Every axis point has one zero coordinate.
  2. sin θ < 0 means y < 0, so the candidates are III and IV.Sine is height.
  3. cos θ < 0 means x < 0, so the candidates are II and III.Cosine is horizontal position.
  4. The shared candidate is III.III appears in both lists.
Answer
Quadrant III.
Check III has both coordinates negative, so it satisfies both conditions.
Rung 3Rung 3: translate reciprocal signs

sec θ > 0 and csc θ < 0. Find the quadrant. First translate the flipped values into coordinate signs.

x = cos θy = sin θright and below
The two reciprocal conditions select IV.
  1. Both secant and cosecant have defined nonzero values, so their coordinate denominators x and y are nonzero. This excludes all axis points.An axis would make one of these reciprocal functions undefined.
  2. sec θ > 0 gives cos θ > 0 and x > 0, so the candidates are I and IV.A reciprocal keeps its nonzero partner's sign.
  3. csc θ < 0 gives sin θ < 0 and y < 0, so the candidates are III and IV.The same reciprocal sign rule applies to cosecant.
  4. Only IV appears in both lists.The terminal point must be right and below at once.
Answer
Quadrant IV.
Check In IV, x > 0 makes sec positive and y < 0 makes csc negative.
Rung 4Rung 4: recognize incomplete information

tan θ < 0 and cot θ < 0. Is one quadrant determined? The question asks whether these clues narrow the location to one choice.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read x from left or right and y from below or above before building the other signs.
  1. tan θ < 0 allows II or IV.Its coordinate signs must differ.
  2. cot θ < 0 also allows II or IV.Reversing top and bottom keeps the quotient's sign when both are nonzero.
  3. Both clues still leave II and IV.These conditions repeat the same sign information rather than removing another candidate.
Answer
Quadrant II or Quadrant IV. One quadrant is not determined.
Check In II, x < 0 and y > 0 give two negative quotients. In IV, x > 0 and y < 0 also give two negative quotients.
Rung 5Rung 5: distinguish an axis from a quadrant

cos θ = 0 and sin θ > 0. Locate the terminal point. One output is zero, so inspect axes.

x = cos θy = sin θ(0, 1)
A zero coordinate puts the point on a boundary.
  1. cos θ = 0 means x = 0, so the point is on the y-axis.Cosine is the x-coordinate.
  2. The unit-circle points with x = 0 are (0, 1) and (0, −1).The circle has radius 1.
  3. sin θ > 0 means y > 0, so choose (0, 1).Only the upper axis point has positive height.
Answer
The positive y-axis, at (0, 1). It is in no quadrant.
Check At (0, 1), cosine is 0 and sine is 1, which meets both conditions.
Rung 6Rung 6: reject contradictory clues

An angle has sin θ > 0, cos θ > 0 and tan θ < 0. Can it exist? The question asks whether all three conditions can hold together.

x = cos θy = sin θx > 0, y > 0
Two positive coordinates force positive tangent.
  1. sin θ > 0 and cos θ > 0 put the point in I.Positive height and positive horizontal position mean upper right.
  2. In I, tan θ = yx is positive.A positive divided by a positive is positive.
  3. That contradicts tan θ < 0.One angle cannot have both a positive and a negative tangent value.
Answer
No angle satisfies all three conditions.
Check The quotient definition rules out the negative tangent directly, even before looking at the memory phrase.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Tangent is negative in Quadrant III because y is negative.
Tangent divides y by x. In III both are negative, so looking only at y loses the second sign.
✓ Instead: tan t > 0 in III, because negativenegative is positive.
✗ Not this: The word Sine in All Students Take Calculus means only sine is positive in II.
The memory device names a function together with its reciprocal. The reciprocal has the same sign.
✓ Instead: Both sin t and csc t are positive in II; the other four are negative.
✗ Not this: An angle on an axis belongs to the neighboring quadrant.
An axis is the boundary between quadrants. One coordinate is 0 there, so the table's strictly positive or negative signs do not apply.
✓ Instead: Use the actual axis point and the six formulas to decide 0, nonzero or undefined.
Tips and tricks
  • Write the x and y signs first. That line can earn useful partial credit and gives you a check on every later sign.
  • For a location from two signs, write two candidate lists and circle only the locations appearing in both.
  • All Students Take Calculus names positive functions, not negative ones.
Trap. A reciprocal keeps its partner's sign. csc follows sin, sec follows cos, and cot follows tan wherever both exist. Remember that the All Students Take Calculus cue excludes axis points.
Keep in mind
  • The sign of an angle and the sign of its sine are different things: 200° is a positive angle, but sin 200° < 0 because its point is below the x-axis.
  • Two sign clues narrow the location: list the quadrants each allows and keep the overlap, so cos t > 0 (I, IV) with cot t < 0 (II, IV) gives Quadrant IV.
  • On an axis one coordinate is 0, so the quadrant table does not apply: at the top point (0, 1), cos t = 0 is neither positive nor negative.
  • A flip keeps the sign: sin t = −0.3 gives csc t = 1−0.3 ≈ −3.33.
Memory hookAll Students Take Calculus: counterclockwise from Quadrant I, the positive ones are All, Sine, Tangent, Cosine, each with its flip (csc, cot, sec).
Flash cards: say the answer out loud, then flip
What is a quadrant?
One of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the upper right.
Signs of (x, y) in Quadrant IV?
(+, −): right and down.
In which quadrants is tan t positive, and why?
I and III: x and y have the same sign there, so yx > 0.
t = 160°. Signs of sin t, cos t and tan t?
  • Quadrant II.
  • sin t: +
  • cos t: −
  • tan t: −
cos t < 0 and cot t < 0. Which quadrant?
II: cos t < 0 allows II and III, and cot t < 0 allows II and IV.
In Quadrant II, is csc t negative because All Students Take Calculus names only Sine there?
No: each named function brings its flip, so sin t and csc t are both positive in II.