if pi/2 is less than or equal to theta is less than or equal to pi and sin of theta equals 5/9 find cos theta and cot theta

Answer 1

The other side of the right triangle is sqrt(56).

The magnitude of the cosine is (1/9) sqrt(56) and since it is in quadrant II, the sign will be -.

The magnitude of the cot (1/5) sqrt(56) and again the sign will be -.

Answer 2

Given:

Π/2 ≤ Θ ≤ Π This says that Θ lies in the 2nd quadrant

90° ≤ Θ ≤ 180° and is < 180°

Sin(Θ) = 5/9 But the Sine in the 2nd quadrant uses the supplement of Θ for trig identities,

so a positive Sin, 9/5, results in the 2nd quadrant and the hypotenuse is positive.

And the Cosecant = 1/ Sine = 9/5

So the Cosine would be derived from the Pythagorean theorem = SquareRoot of (9² - 5²) = √(56) = 2√14

but is a negative x from the origin so Cosine(Θ) =(-2√14)/9.