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Quadrants and Signs

A free Trigonometry lesson from the “Angles, Degrees, and Radians” unit, with a worked example and practice problems including step-by-step solutions.

In standard position, the signs of sin, cos, and tan depend only on which quadrant the terminal side lands in, because sin matches the y-coordinate's sign, cos matches the x-coordinate's sign, and tan is their ratio. The mnemonic ASTC ("All Students Take Calculus") names the one function that is positive in each quadrant, reading counterclockwise from Quadrant I: All in QI, Sine in QII, Tangent in QIII, Cosine in QIV. Knowing one or two signs is enough to pin down a quadrant, and knowing a quadrant tells you the sign of every function at once.

What you'll learn

Why it matters: Whenever an angle or direction is measured (a radar bearing, a rotating crank arm, a phase in an AC circuit), the same coordinates can be positive or negative depending on the quadrant, so engineers use the quadrant signs to know whether a horizontal or vertical component points forward or backward.

Worked example

Problem. An angle of 210 degrees is in standard position. Name its quadrant and state the sign of sin, cos, and tan there.

  1. 210 degrees is between 180 and 270 degrees, so the terminal side is in Quadrant III.
  2. By ASTC, only Tangent is positive in Quadrant III, so tan(210 degrees) is positive while sin and cos are both negative.
  3. Check: in QIII both x and y are negative, so sin (= y sign) and cos (= x sign) are negative, and tan = y/x is positive.

Answer: Quadrant III; sin is negative, cos is negative, tan is positive

Practice problems

1. In Quadrant I, which of sin, cos, and tan are positive?

Show solution
  1. Warm-up: First identify exactly what the question is asking: In Quadrant I, which of sin, cos, and tan are positive?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The 'A' in ASTC stands for All and labels Quadrant I.
  4. So sin, cos, and tan are all positive in Quadrant I.
  5. Check the result by substituting or estimating: the response should match All three (sin, cos, and tan are all positive) and make sense in the original problem.

Answer: All three (sin, cos, and tan are all positive)

2. Which single trig function is positive in Quadrant II?

Choices: sin · cos · tan

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which single trig function is positive in Quadrant II?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The 'S' in ASTC labels Quadrant II and stands for Sine.
  4. So in Quadrant II only sin is positive; cos and tan are negative.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sin

3. In Quadrant III, state the sign (positive or negative) of sin, cos, and tan.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In Quadrant III, state the sign (positive or negative) of sin, cos, and tan.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The 'T' in ASTC labels Quadrant III, so only Tangent is positive.
  4. Therefore sin is negative, cos is negative, and tan is positive.
  5. Check the result by substituting or estimating: the response should match sin negative, cos negative, tan positive and make sense in the original problem.

Answer: sin negative, cos negative, tan positive

4. List the sign of sin, cos, and tan in Quadrant IV.

Show solution
  1. Core Practice: First identify exactly what the question is asking: List the sign of sin, cos, and tan in Quadrant IV.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The 'C' in ASTC labels Quadrant IV, so only Cosine is positive.
  4. Therefore cos is positive while sin and tan are negative.
  5. Check the result by substituting or estimating: the response should match sin negative, cos positive, tan negative and make sense in the original problem.

Answer: sin negative, cos positive, tan negative

5. An angle of 135 degrees is in standard position. Name its quadrant and the sign of cos there.

Show solution
  1. Core Practice: First identify exactly what the question is asking: An angle of 135 degrees is in standard position. Name its quadrant and the sign of cos there.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 135 degrees is between 90 and 180 degrees, so it is in Quadrant II.
  4. In Quadrant II only sin is positive, so cos is negative.
  5. Check the result by substituting or estimating: the response should match Quadrant II; cos is negative and make sense in the original problem.

Answer: Quadrant II; cos is negative

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