Question

If cot ϴ = -3/4 and 90 degrees < ϴ < 180 degrees, find sec ϴ?

Answer 1

`secx=-5/3`

Explanation:

`90^o<theta<180^o` tells us that we're in the second quadrant, where cosine is negative and sine is positive.

So, recall the identity

`1+cot^2theta=csc^2theta`

`cottheta=-3/4, cot^2theta=(-3/4)^2=9/16`

So,

`csc^2theta=16/16+9/16=25/16`

`csctheta=+-sqrt(25/16)=+-5/4`

Knowing that sine is positive and that cosecant is just the reciprocal of sine (and must therefore share the same positive/negative sign), we want the positive answer.

`csctheta=5/4`

Furthermore, `sintheta=1/csctheta=1/(5/4)=4/5`

Now, recalling that `sin^2theta+cos^2theta=1, sin62theta=(4/5)^2=16/25:`

`16/25+cos^2theta=25/25`

`cos^2theta=9/25`

`costheta=+-sqrt(9/25)=+-3/5`

As mentioned above, cosine must be negative in the second quadrant, so we want the negative answer.

`cosx=-3/5`

Finally, recall that `secx=1/cosx`. In this case,

`secx=1/(-3/5)=-5/3`