Question

Question #f46bb

Answer 1

`sec beta=-(3sqrt5)/5`

Explanation:

Using standard identities, `sec beta=1/cos beta=+-1/sqrt(1-sin^2 beta)`
Substituting in `sin beta=-sqrt(2/3)`
`sin^2 beta=4/9`
`1-sin^2 beta=5/9`
`+-sqrt(1-sin^2 beta)=+-sqrt5/3=cos beta`
Since `beta` is in the third quadrant, both `sin beta` and `cos beta` are negative, we need `cos beta=-sqrt5/3`
So `sec beta=1/cos beta=1/(-sqrt5/3)=-3/sqrt5`,
which might also be written, with a rational denominator, as:
`sec beta=-(3sqrt5)/5`