Question

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

Answer 1

Answer:

$-\frac{2}{\sqrt{2}}$ or $-\sqrt{2}$

Step-by-step explanation:

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

In this problem, we're given that $\sin{x}=\frac{\sqrt{2} }{2}$ and $\tan{x}=-1$, so plugging those two values into our equation gives us

$\sec{x}=\dfrac{-1}{\frac{\sqrt{2}}{2}} =-\frac{2}{\sqrt{2}}$

We could leave our solution as $-\frac{2}{\sqrt{2}}$, or we could rationalize the denominator to get a solution of

$-\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=-\frac{2\sqrt{2}}{2}=-\sqrt{2}$