Final answer:
To evaluate ℓ tan(sin⁻¹(-5/13)), identify the angle whose sine is -5/13 and then use a right-angled triangle to find the tangent of that angle. The tangent value is -5/12.
Step-by-step explanation:
Evaluating ℓ tan(sin⁻¹(-5/13)) involves interpreting the expression to mean the tangent of the angle whose sine is -5/13. First, let's find the angle whose sine is -5/13. The inverse sine function can be used here, giving us an angle whose sine is -5/13.
Let θ = sin⁻¹(-5/13)
Now, we consider a right-angled triangle where the opposite side is -5 and the hypotenuse is 13. The adjacent side can be found using the Pythagorean theorem: adjacent = √(hypotenuse² - opposite²) = √(13² - (-5)²) = √(169 - 25) = √144 = 12.
Therefore, tan(θ) = opposite/adjacent = -5/12.
The negative sign indicates that the angle θ is in either the third or fourth quadrant, but since the range of sin⁻¹ is [-π/2, π/2] or [-90°, 90°], the angle is in the fourth quadrant where sine is negative and cosine is positive, making tangent negative as well.
The value of tan(sin⁻¹(-5/13)) is thus -5/12.