Use the given information to find the values of the remaining five trigonometric functions

Question

cot(x) = 1/2, pi < x < 3pi/2

(Please add an explanation of how to find each for future reference. Thank you!!)

Answer 1

pi < x < 3pi/2 tells us angle x is in the third quadrant.

If cot x = 1/2

tan x = 2/1 because tan x is reciprocal of cot x

So tan x = 2

Since cot = adj/opp, adj = 1 and opp = 2 we can use Pythagorean theorem to find the hypotenuse.

1² + 2² = hyp²

5 = hyp^2√

√ 5 = hyp

now sin = opp/hyp

sin x = 2/√ 5....rationalize to get (2√ 5) / 5 since sin is negative in 3rd quadrant sin x = - (2√ 5) / 5

csc x is reciprocal of sin and also negative in 3rd quadrant so csc = - √ 5/ 2

See if you can find cos x and sec x in same manner as above

Answer 2

Use the given information to find the values of the remaining five trigonometric functions

cot(x) = 1/2, pi < x < 3pi/2

cot(x) = 1/2,

cot x = 1/ tan x

1/2 = 1/ tan x

tan x = 2

cot(x)=adjacent / opposite = 1/2

 

Hypotenuse=√(opposite²+adjacent²)

= √ ( 2² + 1² )

= √ 5

sin x = opp/ hyp = 2/ √ 5

but pi < x < 3pi/2

sin value are negative in third quadrant

sin x = - 2/ √ 5

cos x = adj / hyp = 1/ √ 5

cos value are negative in third quadrant

cos x =-1/ √ 5

sec x = 1/ cos x = 1/ -1/ √ 5 = -√ 5

sec x = -√ 5

cosec x =1/ sin x =1/ -2/ √ 5 = -√ 5 / 2

cosec x = -√ 5 / 2