Find tan(arcsin -7/25)

Could someone explain how to find the answer to that? It would really help me alot :)

arcsin(-7/25) means the angle (θ) whose sine is -7/25. This means implies the equation:

sin(θ) = -7/25

We know that, sin²(θ) + cos²(θ) = 1, so plug in (-7/25)² for sin²(θ):

(-7/25)² + cos²(θ) = 1

Simplify the square:

49/625 + cos²(θ) = 1

Subtract 49/625 from both sides:

cos²(θ) = 1 - 49/625

Make 1 = 625/625:

cos²(θ) = 625/625 - 49/625

cos²(θ) = 576/625

The square root of 576 just happens to be ±24. (not really a surprise given that 7, 24, 25 is a right triangle). Therefore, cos(θ) = ±24/25

We know that, tan(θ) = sin(θ)/cos(θ). Because cos(θ) has two possible values, the tangent has two possible values:

tan(θ) = {-7/25}/{24/25} = -7/24

or

tan(θ) = {-7/25}/{-24/25} = -7/-24 = 7/24

angle is in the 4th quadrant with opposite = -7 and hypotenuse = 25
adjacent side is therefore 24 (by Pythagoras)
tangent = -7/24