how to find this trigonometric value

Question

Find Cot(x) when Sec(x) = 9/4 and x in QIV

Answer 1

Given: Sec(x) = 9/4 = Hypotenuse/Adjacent x represents angle in Quad. IV

Find: Cot(x) = Adjacent/Opposite = ?

Using the Pythagorean Theorem we find the unknown leg of the right triangle formed in quadrant IV

a² + b² = c² a - represents our adjacent leg, b - represents our unknown leg (opposite), c -represents the hypotenuse

4² + b² = 9²

16 + b² = 81

b² = 81 - 16 subtract 16 from both sides of the equation

b² = 65

b = ±√65

b = –√65 the negative solution is the one we use for the angle x formed in quadrant IV

Cot(x) = Adj/Opp = –4 / √65

= –4(√65) / (√65)(√65)

Cot(x) = –4(√65) / 65