Question

If sin(x) = Negative one-third and cos(x) > 0, what is tan(2x)?.

Answer 1

To find the value of tan(2x), use the double angle formula and substitute the given values of sin(x) and cos(x). The value of tan(2x) is -2/5.

Step-by-step explanation:

To find the value of tan(2x), we need to use the double angle formula for tangent. The formula is tan(2x) = (2tan(x))/(1-tan^2(x)).

Given that sin(x) = -1/3 and cos(x) > 0, we can find the value of tan(x) by using the fact that tan(x) = sin(x)/cos(x).

Using the given information, we have tan(x) = (-1/3)/cos(x). Since cos(x) > 0, we know that cosine is positive in the first and fourth quadrants. Therefore, tan(x) < 0 because sin(x) < 0 (as given).

So, by substituting tan(x) = -1/3 into the double angle formula, we get tan(2x) = (2(-1/3))/(1-(-1/3)^2).

Simplifying the expression, we have tan(2x) = -2/5.

Answer 2

The correct answer is (a)
`$-(4 √(2))/(7)$`.

We can use the double-angle identity for tangent to solve this problem:

`\tan (2 x)=(2 \tan (x))/(1-\tan ^2(x))`

First, let's find
`$\tan (x)$` using the given information:

`\tan (x)=(\sin (x))/(\cos (x))=(-(1)/(3))/(\cos (x))`

Since
`$\cos (x) > 0$`, we know that
`$\tan (x)$` is negative.

Therefore,
`$\tan (x)=-(√(2))/(3)$`

Now we can plug
`$\tan (x)=-(√(2))/(3)$` into the double-angle identity:

`\tan (2 x)=(2\left(-(√(2))/(3)\right))/(1-\left(-(√(2))/(3)\right)^2)=(-(2 √(2))/(3))/((7)/(9))=-(4 √(2))/(7)`

Complete Question:

If
`$\sin (x)=-(1)/(3)$` and
`$\cos (x) > 0$`, what is
`$\tan (2 x)` ?

a.
`& -(4 √(2))/(7)`

b.
`-(4 √(2))/(9)`

c.
`(4 √(2))/(9)`

d.
`(4 √(2))/(7)`