SOLUTION: What is tan theta when sin theta equals 1/4 and theta is in quadrant one

Question 593060: What is tan theta when sin theta equals 1/4 and theta is in quadrant one
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Problems like these can be broken down into two parts: Finding the desired ratio and determining the correct sign (positive or negative) for that ratio. I usually find the ration first but it's OK to find the sign first.

Finding the desired ratio:
1. If the given ratio is not written as a fraction, rewrite it as a fraction. For example, if you were given cos(x) = 0.6, then you should rewrite it as cos(x) = 6/10. NOTE: If the given ratio, as a fraction, is recognizable as the ratio for a special angle then you should be able to find the desired ratio without having to do any of the rest of this procedure. Otherwise...
2. Draw a right triangle and name one of the acute angles (it doesn't matter which) as the angle in question. In your problem label the angle as theta.
3. Take the numerator and denominator of the fraction and make them the lengths of the two sides appropriate for the given ratio. In your problem your are give the sin ratio. Since sin is opposite over hypotenuse and since the ratio is 1/4, make the opposite side to theta be 1 and the hypotenuse a 4. (NOTE: If the given ratio is negative, just ignore the negative for now. Sides of triangles should be positive.)
4. Find the desired ratio. This may require using the Pythagorean Theorem to find the third side. In your problem the desired ratio is tan. Since tan is opposite over adjacent and since we do not yet know the side adjacent to theta, we will have to use the Pythagorean Theorem:
  (using "a" for the adjacent side)
  Solving...
  
  Now we can construct the desired ratio:
  
  Since we don't usually leave square roots in denominators, we will rationalize the denominator:
Determining the sign:
1. Determine the quadrant(s) where the angle in question may terminate. Your problem is easy in this regard because your have been told that theta terminates in quadrant one.
2. Determine the sign for the desired ratio in that quadrant. Since theta is in quadrant one and since tan (and other other ratios) are positive in that quadrant, then the desired ratio should be positive. (NOTE: If theta had been in quadrant two (where sin could still be +1/4) instead then we would make our answer negative because tan is negative in the second quadrant.)

So our final, rationalized answer is:

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