Quarry School

Signs by quadrant: find the quadrant first

Explain it like I am five

Picture a city map cut by two streets into four neighborhoods, the quadrants, numbered I to IV counterclockwise from the upper right. A point's x says right (+) or left (−), and its y says up (+) or down (−). Since r is always positive, sin θ = yr takes the sign of y, cos θ = xr the sign of x, and tan θ = yx is positive when x and y match.

Try sin θ = −725 with cos θ < 0. Clue one, sine negative: down, so III or IV. Clue two, cosine negative: left, so II or III. The overlap is III. Size next: cos2θ = 1 − 49625 = 576625, and 576625 = 2425. Sign last: III is on the left, so cos θ = −2425, and tan θ = (−725) ÷ (−2425) = 724, positive, as III requires.

A root sign, as in 9 = 3, means the nonnegative (zero or positive) principal square root, so it gives the size. The equation cos2θ = 576625 allows 2425 or −2425, and the quadrant picks one.

In plain words

Picture a city map split into four neighborhoods by an east to west street and a north to south street. A point's x-coordinate tells right or left; its y-coordinate tells up or down. Those directions decide the signs of the trig values because r, the distance from the center, is positive. Each neighborhood is a Quadrant. Upper right is I, upper left II, lower left III and lower right IV. Sine follows up or down. Cosine follows right or left. Tangent is positive when both directions have matching signs. Before finding a missing value, locate the neighborhood. The square root tells its size; the neighborhood tells whether to place a minus sign in front.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
r is always positive, so sine follows the sign of y, cosine follows the sign of x, and tangent is positive where they match.
Reminder
  • Squares of negatives. (−45)2 = 1625 because a negative times a negative is positive.
  • Subtracting from one. 1 − 449 = 4949 − 449 = 4549. Match the denominators first.
  • Principal square root. 1625 = 45, while solving z2 = 1625 gives z = ±45.
  • Rationalizing. 235 × 55 = 2515; the multiplying fraction equals 1.
Why it works. sin θ = yr has the sign of y because r > 0. cos θ = xr has the sign of x. tan θ = yx is positive when x and y have matching signs and negative when they differ. A reciprocal of a nonzero number keeps its sign, so csc, sec and cot copy their partners inside a quadrant. Squaring loses signs: both 45 and −45 square to 1625. An identity gives the magnitude of a missing sine or cosine; the quadrant restores its sign.
RuleInside Quadrants I, II, III, IV, the positive functions are respectively: all six; sin and csc; tan and cot; cos and sec. Every remaining function is negative.
sin2θ + cos2θ = 1 gives the missing magnitude. The quadrant chooses the sign of sin θ or cos θ in front of its nonnegative square root.
The same idea, five ways
Say it

Find the quadrant first. A sign clue is a direction clue.

Write it

The signs of sine and cosine reveal the signs of y and x; tangent compares those signs.

In math
  • sin θ > 0 means y > 0
  • cos θ < 0 means x < 0
  • tan θ > 0 means x and y have the same sign
  • cos θ = ±1−sin2θ, with the quadrant selecting the sign
Like

Use two directions, left and up, to locate one neighborhood on a map.

See it
QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
r is always positive, so sine follows the sign of y, cosine follows the sign of x, and tangent is positive where they match.
The same idea, other ways
As directions on a map

Sine says above or below, cosine says right or left. In Quadrant II, left means x < 0 and up means y > 0. So sine and cosecant are positive, while cosine, secant, tangent and cotangent are negative.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Read signs from the point's directions instead of guessing them.
As signed division

For a point in Quadrant III, take x = −2 and y = −5. Sine and cosine are negative because r is positive. Tangent is −5−2 = 52, positive because both signs match.

xyθx = −2y = −5rP(−2, −5)Ox = −2 (2 left) y = −5 (5 down) r = √29.0 ≈ 5.4red ray from O through P (and beyond) = the terminal side
Two negative coordinates make a positive tangent.
As a memory sentence

All Students Take Calculus lists the positive families in counterclockwise order: All in I, Sine in II, Tangent in III, Cosine in IV. Include each named function's reciprocal partner. Use this for signs inside quadrants; the axes need their own point calculation.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
All, Sine, Tangent, Cosine names the positive families in I, II, III, IV.
Quadrantxysin θ and csc θcos θ and sec θtan θ and cot θ
I+++++
II−++−−
III−−−−+
IV+−−+−
.1Combine sign clues

One clue often leaves two neighborhoods on the map. A second clue narrows the search. List both sets and keep their overlap. A positive cosecant means a positive sine because a nonzero number and its reciprocal share a sign. A negative tangent means the two coordinates have opposite signs.

  • sin and csc positive: I or II. Negative: III or IV.
  • cos and sec positive: I or IV. Negative: II or III.
  • tan and cot positive: I or III. Negative: II or IV.
  • All Students Take Calculus names the positive families; their reciprocal partners share those signs.
QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Each clue removes quadrants where it cannot hold.
Worked exampleTwo directions choose one quadrant

cos β < 0 and csc β > 0. Find the quadrant. In words, find the region where both clues can hold.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Left and up overlap in Quadrant II.
  1. cos β < 0 gives II or III.Cosine has the sign of x, so the point is left of the y-axis.
  2. csc β > 0 means sin β > 0, giving I or II.A nonzero number and its reciprocal share a sign.
  3. Only II is in both lists.The point must be both left and up.
Answer
Quadrant II
Check In II, cosine is negative and sine and cosecant are positive, matching both clues.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Negative tangent always means Quadrant IV.
Opposite coordinate signs occur in both II and IV.
✓ Instead: Use another sign clue to distinguish II from IV.
Tips and tricks
  • Draw two short lists and circle the common quadrant.
.2Recover the missing magnitude and sign

Knowing one coordinate of a point whose distance from the origin is 1 is like knowing one leg of a triangle whose longest side is 1. The squares must add to 1, so subtract the known square to find the missing square. Take a root to find the size. Then use the quadrant to decide the sign. These are two separate decisions.

  • cos2θ = 1 − sin2θ and sin2θ = 1 − cos2θ.
  • a is the nonnegative principal root for a ≥ 0.
  • A positive root describes size; the quadrant supplies the sign of sine or cosine.
x = cos θy = sin θ([[−4|5]], [[−3|5]])
The triangle's sizes are positive, but the Quadrant III coordinates are negative.
Worked exampleRecover sine in Quadrant III

cos β = −45 and β is in III. Find sin β and tan β. In words, recover the other coordinate and then its ratio to cosine.

x = cos θy = sin θ([[−4|5]], [[−3|5]])
Both coordinates are negative, so their ratio is positive.
  1. sin2β = 1 − (−45)2 = 2525 − 1625 = 925.The Pythagorean identity isolates the square of the missing sine.
  2. sin β = −925 = −35.The root gives size 35, and Quadrant III requires a negative sine.
  3. tan β = (−35) ÷ (−45) = 34.The equal denominators cancel, and matching negative signs give a positive ratio.
Answer
  • sin β = −35
  • tan β = 34
Check 925 + 1625 = 1, and tangent is positive in III.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Solving sin2β = 925 gives only 35.
Both signs have the same square. The principal root alone does not identify sin β.
✓ Instead: The equation gives ±35; Quadrant III selects −35.
Tips and tricks
  • Write size first, then sign, to see where each decision came from.
Strategy: step by step
  1. For each clue, list every quadrant where it is true. A zero value points to an axis instead.
  2. Keep only the quadrants shared by every clue. If more than one remains, report that the information does not determine one quadrant.
  3. Use sin2θ + cos2θ = 1 to isolate the square of the missing partner.
  4. Take the nonnegative square root for its magnitude, then attach the sign required by the quadrant.
  5. Compute tan = sinθcosθ, cot = cosθsinθ, sec = 1cosθ and csc = 1sinθ, checking denominators.
  6. Check the squared identity and every sign against the clues.
Strategy
Find a quadrant and recover the missing functions
1
Is a given sine or cosine zero?
YesUse the axis points, including undefined values.
NoList its possible quadrants.
↓
2
Do the clues leave exactly one common quadrant?
YesUse that quadrant to choose each missing sign.
NoKeep all surviving possibilities; no common quadrant means the clues conflict.
↓
3
Is sine or cosine given?
YesRecover the partner with sin2θ + cos2θ = 1.
NoTurn a reciprocal into its partner, or build signed x and y from tangent.
  1. For each clue, list every quadrant where it is true. A zero value points to an axis instead.
  2. Keep only the quadrants shared by every clue. If more than one remains, report that the information does not determine one quadrant.
  3. Use sin2θ + cos2θ = 1 to isolate the square of the missing partner.
  4. Take the nonnegative square root for its magnitude, then attach the sign required by the quadrant.
  5. Compute tan = sinθcosθ, cot = cosθsinθ, sec = 1cosθ and csc = 1sinθ, checking denominators.
  6. Check the squared identity and every sign against the clues.
Worked exampleUse two sign clues before taking a root

sin θ = 27 and tan θ < 0. Find the other five functions. In words, decide where the terminal side is, then recover the missing ratios.

x = cos θy = sin θ(−[[3√{5}|7]], [[2|7]])
The point has positive y and negative x, so the required sine and tangent signs meet in Quadrant II.
x = cos θy = sin θ(−[[3√{5}|7]], [[2|7]])
The point has positive y and negative x, so the required sine and tangent signs meet in Quadrant II.
input quadrantoutput xI+II−III−IV+
Reference table: read the x entry directly under its quadrant label.
input quadrantoutput yI+II+III−IV−
Reference table: read the y entry directly under its quadrant label.
input quadrantoutput sin θ and csc θI+II+III−IV−
Reference table: read the sin θ and csc θ entry directly under its quadrant label.
input quadrantoutput cos θ and sec θI+II−III−IV+
Reference table: read the cos θ and sec θ entry directly under its quadrant label.
input quadrantoutput tan θ and cot θI+II−III+IV−
Reference table: read the tan θ and cot θ entry directly under its quadrant label.
  1. sin θ > 0 allows I or II; tan θ < 0 allows II or IV. Their common quadrant is II.Only Quadrant II satisfies both sign clues, so cosine must be negative.
  2. cos2θ = 1 − (27)2 = 4949 − 449 = 4549.This isolates the square of the unknown cosine.
  3. cos θ = −4549 = −357.45 = 35; choose the negative root for Quadrant II.
  4. tan θ = 27 ÷ (−357) = −235 = −2515.The 7s cancel; multiply top and bottom by 5.
  5. cot θ = −352; sec θ = −735 = −7515; csc θ = 72.Take reciprocals of nonzero tangent, cosine and sine.
Answer
  • Quadrant II
  • cos θ = −357
  • tan θ = −2515
  • cot θ = −352
  • sec θ = −7515
  • csc θ = 72
Check sin2θ + cos2θ = 449 + 4549 = 1. Only sine and cosecant are positive, as required in Quadrant II.
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 without arithmetic

sin α < 0 and cos α > 0. Find the quadrant. In words, locate the point below and to the right.

QI (+, +)All positiveQII (−, +)Sine, csc positiveQIII (−, −)Tangent, cot positiveQIV (+, −)Cosine, sec positive
Right and down is Quadrant IV.
  1. sin α < 0 permits III or IV; cos α > 0 permits I or IV.Sine follows y, and cosine follows x.
  2. Only IV is in both lists.Both clues must be true together.
Answer
Quadrant IV
Check In IV the point is right and down, which gives positive x and negative y.
Rung 2Rung 2: a missing partner with whole-number roots

sin β = −45 and tan β > 0. Find cos β and tan β. In words, choose the quadrant and recover cosine.

x = cos θy = sin θ([[−3|5]], [[−4|5]])
The negative sine and positive tangent place the point in III.
  1. Negative sine allows III or IV; positive tangent allows I or III. Choose III.III is their only overlap, so cosine is negative.
  2. cos2β = 1 − 1625 = 925, so cos β = −35.The squared identity gives magnitude 35, and III gives the sign.
  3. tan β = (−45) ÷ (−35) = 43.Equal bottoms cancel; two negative signs give a positive quotient.
Answer
  • Quadrant III
  • cos β = −35
  • tan β = 43
Check 1625 + 925 = 1, and the positive tangent matches its clue.
Rung 3Rung 3: all six with a radical

sin θ = 27 and tan θ < 0. Find the other five functions. In words, use the signs before recovering a radical cosine.

x = cos θy = sin θ(−[[3√{5}|7]], [[2|7]])
Positive y and negative x give the required signs.
  1. Positive sine allows I or II; negative tangent allows II or IV. Choose II.II is their common quadrant.
  2. cos2θ = 1 − 449 = 4549, so cos θ = −357.45 = 35, and cosine is negative in II.
  3. tan θ = −235 = −2515 and cot θ = −352.Divide sine by cosine, rationalize tangent, and reverse that nonzero ratio.
  4. sec θ = −735 = −7515 and csc θ = 72.Take the reciprocals of cosine and sine, rationalizing the root denominator.
Answer
  • cos θ = −357
  • tan θ = −2515
  • cot θ = −352
  • sec θ = −7515
  • csc θ = 72
Check The squares 449 and 4549 add to 1; only sine and cosecant are positive.
Rung 4Rung 4: start from tangent

tan γ = 3 and cos γ < 0. Find the other five functions. In words, choose a signed point with the required y-to-x ratio.

xyθx = −1y = −3rP(−1, −3)Ox = −1 (1 left) y = −3 (3 down) r = √10.0 ≈ 3.2red ray from O through P (and beyond) = the terminal side
Both coordinates are negative and yx = 3.
  1. Positive tangent allows I or III; negative cosine allows II or III. Choose III.Both clues hold in III, where x and y are negative.
  2. Use x = −1, y = −3, so yx = 3. Then r = 1+9 = 10.A point on that ray can have any positive scale; these coordinates give the required ratio.
  3. sin γ = −310 = −31010 and cos γ = −110 = −1010.Use yr and xr, then rationalize.
  4. cot γ = 13, sec γ = −10, csc γ = −103.Use xy, rx and ry.
Answer
  • sin γ = −31010
  • cos γ = −1010
  • cot γ = 13
  • sec γ = −10
  • csc γ = −103
Check tan2γ + 1 = 9 + 1 = 10 = sec2γ, and only tangent and cotangent are positive.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: sin θ = 27 means θ is in Quadrant I.
Positive sine means y > 0, which holds in both I and II.
✓ Instead: Use the second clue tan θ < 0 to choose II.
✗ Not this: cos θ = 4549 = 357 in the main example.
That positive value contradicts Quadrant II, where x < 0.
✓ Instead: cos θ = −357. The root supplies magnitude; the minus sign comes from the quadrant.
Tips and tricks
  • Know All Students Take Calculus for the signs, then rebuild it from x and y when needed.
  • Write the chosen quadrant before touching a square root. It controls the sign of the missing value.
  • In Quadrant III, tangent and cotangent are positive because both coordinate signs match; sine, cosine, cosecant and secant remain negative.
  • Do not use the quadrant sign chart for axis points. An axis value can be zero or undefined.
  • Read reference tables by columns: keep the input label and its output in the same vertical column.
Trap. Taking the positive square root out of habit. The principal square root, written a, is nonnegative, including 0 = 0. A missing sine or cosine can still be negative: choose its sign from the quadrant. When a sign clue leaves two possible quadrants, keep both instead of guessing.
Keep in mind
  • An angle's sign and its sine's sign are different things: 200° is a positive angle, yet sin 200° < 0 because its point sits below the x-axis.
  • The root sign gives only the nonnegative root, so 949 = 37, while x2 = 949 has two answers, x = 37 or x = −37.
  • Each reciprocal copies its partner's sign: cos θ = −2425 gives sec θ = −2524.
  • With two clues, list each clue's two quadrants and keep the overlap: sec θ > 0 allows I or IV, csc θ < 0 allows III or IV, so θ is in IV.
Memory hookAll Students Take Calculus: counterclockwise from Quadrant I, All six are positive, then Sine (and csc), then Tangent (and cot), then Cosine (and 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.
What is the principal square root?
The nonnegative root, the one the root sign means: 481 = 29, not −29.
In All Students Take Calculus, which functions are positive in Quadrant III?
Tangent and cotangent (the T).
cot θ > 0 and sin θ > 0. Which quadrant is θ in?
I. cot > 0 allows I or III, sin > 0 allows I or II, and the overlap is I.
200° is past 180°. Is 200° a negative angle?
No. 200° is positive, a counterclockwise turn. Its sine is negative because its point is below the x-axis.