Math Snap

STEP 1

What is this asking?
If the sine of an angle is $-1/3$ and its cosine is negative, what's the tangent of that angle?
Watch out!
Remember $\sin$ is negative in the third and fourth quadrants, while $\cos$ is negative in the second and third quadrants.
Both are negative only in the third quadrant!

STEP 2

1. Find Cosine
2. Calculate Tangent

STEP 3

We know that $\sin \theta = -1/3$ and $\cos \theta < 0$.
Let's visualize this on the unit circle!
Since both sine (the y-coordinate) and cosine (the x-coordinate) are negative, our angle $\theta$ must be in the third quadrant!

STEP 4

Let's use the Pythagorean identity:
$$\sin^2 \theta + \cos^2 \theta = 1$$ We're doing this because it connects sine and cosine, and we know one and want to find the other!

STEP 5

Substitute the known value of $\sin \theta$:
$$(-1/3)^2 + \cos^2 \theta = 1$$

STEP 6

Simplify and solve for $\cos \theta$:
$$1/9 + \cos^2 \theta = 1$$ $$\cos^2 \theta = 1 - 1/9$$$$\cos^2 \theta = 8/9$$$$\cos \theta = \pm \sqrt{8/9} = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3}$$Since we know $\cos \theta < 0$ (from the problem statement!), we take the negative solution:
$$\cos \theta = -\frac{2\sqrt{2}}{3}$$

STEP 7

Now, let's recall the definition of tangent:
$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ We're using this because we want to find $\tan \theta$, and we know both $\sin \theta$ and $\cos \theta$!

STEP 8

Substitute the values we found:
$$\tan \theta = \frac{-1/3}{-2\sqrt{2}/3}$$

STEP 9

Simplify the fraction by multiplying the numerator and denominator by $-3$ (to divide to one):
$$\tan \theta = \frac{1}{2\sqrt{2}}$$

STEP 10

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$:
$$\tan \theta = \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2 \cdot 2} = \frac{\sqrt{2}}{4} = \frac{1}{\sqrt{8}}$$ We do this to avoid having a square root in the denominator, which is often preferred for cleaner presentation.

The exact value of $\tan \theta$ is $\frac{1}{\sqrt{8}}$, which corresponds to answer choice b.