Question

Find tan if sin theta= 1/4 and falls in Quadrant 2.

Answer 1

Here we need to know SOH CAH TOA. So first thing first we know tan is negative in quadrant 2 because the x value is negative in quadrant to and since tan is opposite/adjacent (y/x) it will be negative. so now think of a standard graph with 4 quadrants. Now imagine or draw a diagonal line in quadrant 2. Now draw a dotted line straight down from the end of that point to the x axis creating a right triangle. theta is the angle closest to the origin of the graph. SO we know sinΘ= 1/4 and sin= opposite/hypotenuse. So the opposite line (the vertical one we drew down to touch the x axis) is 1 and the hypotenuse is 4. Well to find tan we need the adjacent line (horizontal) which we will find using the Pythagorean theorem
a² + b² = c²
1² + b² = 4²
1 + b² = 16
b² = 15
b =√15        so the adjacent side of our triangle is -√15 because remember we are in the 2nd quadrant which is a negative x. SO since tan= opposite/adjacent:
tan= 1/ -√15  or -1/√15       this could be your answer unless your teacher doesn't want radicals on the bottom of a fraction. to remove simply multiply the top and bottom by the radical.
-1/√15
(-1 × √15) / (√15 × √15)
-√15/15            is ur final answer

Answer 2

If sinα=1/4 and it is in q2 it means that the x coordinate is negative.

The triangle has a height of one and a hypotenuse of 4 so:

x=-√(4^2-1)

x=-√15

so tanα=-1/√15

tanα=-√(1/15)

tanα≈ -0.258