If cot (theta) = 2.5 and theta is in quadrant 3, what is the value of sin (theta)?

Answer 1

Since cotangent is the reciprocal function of tangent,

cot(Θ) = 2.5 = 5/2

then

tan(Θ) = 2/5 (although tangent is positive in quadrant 1 and 3 - but we were given Θ is in quadrant 3)

Both x and y coordinates in Q3 are negative so sine (sim), cosine (cos) and their reciprocal function - cosecant (csc) and secant (sec) are ALL negative.

Sine is the ratio of the opposite side length to the length of the hypotenuse. First, we have to determine the length of the hypotenuse. Fortunately, we know the ratio of the other sides = 2/5. If we use these as lengths we get:

c² = a² + b²

c² = (2)² + (5)²

c² = 29

c = √29

Recalling that the opposite length is 2 and using the hypotenuse is √29 we gte the value for sine as:

sin(Θ) = -2/(√29) = -(2√29)/29 or ~ -0.3716

Note Θ = reference angle plus 180º

(Reference angle is angle in first quadrant where sine is 0.3716 (which is 21.8º))

So Θ = 21.8º + 180º = 201.8º

Answer 2

To solve this draw a line segment in the xy-plane from origin to the point (–2.5,–1).

This corresponds to the angle θ in the third quadrant because x and y are both negative and

cot (θ) = adjacent/opposite = x/y = –2.5/–1 = 2.5

The length of this segment (using the Pythagorean theorem) is given by

r = √(x² + y²) = √((–2.5)² + (–1)²) = √((6.25 + 1) ≈ 2.693

Now we can calculate sin (θ) = opposite/hypotenuse = y/r ≈ –1/2.693. So we have

sin (θ) ≈ –0.371