Final answer:
The exact value of tan(arcsin(-√2/2)) is -1. This is determined by understanding the relationship between the sine and tangent functions in a right triangle, specifically for the angle with a sine of -√2/2, which corresponds to an angle of -45°.
Step-by-step explanation:
To find the exact value of tan(arcsin(-√2/2)), we need to consider the properties of a right triangle where one of the angles has a sine of -√2/2. If we call this angle θ, then sin(θ) = -√2/2.
Since the sine of an angle in a right triangle is the opposite side over the hypotenuse, and the angle whose sine is -√2/2 is -45° (or 315° in standard position), we can use a reference right triangle with sides of √2 (opposite), √2 (adjacent), and 2 (hypotenuse).
The tangent of an angle in a right triangle is the length of the opposite side divided by the length of the adjacent side. So, tan(θ) would be (-√2)/(√2), which simplifies to -1. Therefore, the exact value of tan(arcsin(-√2/2)) is -1.