Question

If sin x= (sqrt 2)/2 and tan x=-1 what is sec x

Answer 1

`-(2)/(√(2))` or
`-√(2)`

Explanation:

`\sec{x}` is the reciprocal of
`cos(x)`, or in symbols:
`\sec{x}=(1)/(cos(x))`.
`tan(x)` is also the ratio of
`sin(x)` to
`cos(x)`, and we can multiply both sides of the equation
`tan(x)=(sin(x))/(cos(x))` by
`(1)/(sin(x))` to get the equation
`(tan(x))/(sin(x))=(1)/(cos(x))`. Of course, this is just the definition of
`\sec{x}`, so we can rewrite this fact as
`\sec{x}=(tan(x))/(sin(x))`.

In this problem, we're given that
`sin(x)=(√(2) )/(2)` and
`tan(x)=-1`, so plugging those two values into our equation gives us

`\sec{x}=(-1)/((√(2))/(2)) =-(2)/(√(2))`

We could leave our solution as
`-(2)/(√(2))`, or we could rationalize the denominator to get a solution of

`-(2)/(√(2))\cdot(√(2))/(√(2))=-(2√(2))/(2)=-√(2)`