If cos t = negative sqrt 11/4 and if the terminal point determined by t is in quadrant II find Cot t sec t + csc

Question

really hard!!

Answer 1

Hi Tetyana,

if cos t = -sqrt(11)/4, then the hypotenuse of the triangle is 4 and the adjacent side is sqrt(11). 

Use the Pythagorean theorem to find the opposite side length

4^2 = sqrt(11)^2+ x^2

x = sqrt(5) units is the length of the opposite side

Now we know the side lengths of the triangle, we can find all the trigonometric functions value. Just be careful with signs because we are in quadrant II!

The expression we are given is

Cot t sec t + csc t

 

cot t = cos t/sin t

sin t = sqrt(5)/4

cot t = (-sqrt(11)/4)/(sqrt(5)/4)

= -sqrt(11)/sqrt(5)

sec t = 1/cos t

= 1/(-sqrt(11)/4)

= -4/sqrt(11)

csc t = 1/sin t

= 1/(sqrt(5)/4)

= 4/(sqrt(5)

Now, we know the values of all the functions in the expression. 

Cot t sec t + csc t = (-√11/√5)(-4/√11) + (4/√5)

= (4/√5) + (4/√5)

= 8/√5

= 8√5 / 5