What is the sign of the following?

Question

cot (θ + 180°), given that θ is in the interval (270°, 360°).

Answer 1

What is the sign of: cot (θ + 180°), given that θ is in the interval (270°, 360°)

Remember: ASTC – All Students Take Calculus (Means: Engineering Students)

* A (All) – In 1st Quadrant: All sin, cos, tan and their reciprocal csc, sec, cot Positive

* S (sin) – In 2nd Quadrant: Only sin and its reciprocal csc Positive (The rest Negative)

* T (tan) – In 3rd Quadrant: Only tan and its reciprocal cot Positive (The rest Negative)

* C (tan) – In 4th Quadrant: Only cos and its reciprocal sec Positive (The rest Negative)

Given: θ is in fourth quadrant - in the interval (270°, 360°).

Then: θ + 180° is in second quadrant. In 2nd Quadrant, tan and cot are Negative.

Therefore, the sign of cot (θ + 180°) in 2nd Quadrant are Negative

Answer 2

given: 270° ≤ θ ≤ 360°

270° + 180° ≤ (θ + 180°) ≤ 360° + 180°

450°≤ (θ + 180°) ≤ 540°

after 360° a cycle repeats, so we can look at:

450° - 360° ≤ (θ + 180°) ≤ 540° - 360°

90°≤ (θ + 180°) ≤ 180°

This is in the 2nd quadrant where COT (reciprocal of TAN) is negative (remember ASTC)