Question

How do you use fundamental identities to find the values of the trigonometric values given θ be an angle in quadrant III, such that cot θ = (√11)/(5)?

Answer 1

The given `cot theta=sqrt(11)/5`,
the answers `tan theta=5/sqrt(11)`, `" "sin theta=-5/6`, `" "cos theta=-sqrt(11)/6`,`" "csc theta=-6/5`, `" "sec theta=(-6sqrt11)/11`

Explanation:

the given `cot theta` at the third quadrant x is negative and y is negative.

The formulas:
`tan theta=1/cot theta`
`csc theta=+-sqrt(cot^2 theta+1)`
`sin theta=1/csc theta=1/(+-sqrt(cot^2 theta+1))`
`cos theta=sin theta*cot theta=cot theta/(+-sqrt(cot^2 theta+1))`
`sec theta=1/cos theta=(+-sqrt(cot^2 theta+1))/cot theta`

Compute the values of the other remaining trigonometric functions

Start with `cot theta=(-sqrt(11))/(-5)`
`tan theta=1/cot theta=1/((-sqrt(11))/(-5))=5/sqrt(11)=(5sqrt11)/11`

`csc theta=+-sqrt(cot^2 theta+1)=+-sqrt(((-sqrt(11))/(-5))^2+1)=+-sqrt(11/25+1)=-6/5`

`sin theta=1/csc theta=1/(+-sqrt(cot^2 theta+1))=1/(+-sqrt(((-sqrt(11))/(-5))^2+1))=1/(+-sqrt((11/25+1)))=-5/6`

`cos theta=sin theta*cot theta=cot theta/(+-sqrt(cot^2 theta+1))((-sqrt(11))/(-5))/(+-sqrt(((-sqrt(11))/(-5))^2+1))=((-sqrt(11))/(-5))/(+-6/5)=-sqrt(11)/6`

`sec theta=1/cos theta=(+-sqrt(cot^2 theta+1))/cot theta=+-sqrt(((-sqrt(11))/(-5))^2+1)/((-sqrt(11))/(-5))=(-6/5)/((-sqrt(11))/(-5))=(-6)/sqrt(11)=(-6sqrt11)/11`

God bless....I hope the explanation si useful.