Final answer:
To evaluate cos(sin^-1(6/x)), envision a right-angled triangle with the opposite side of length 6 and the hypotenuse of length x. Use the Pythagorean theorem to determine the adjacent side, and then calculate the cosine as the ratio of this adjacent side to x. Ensure that x is at least 6 for a valid triangle.
Step-by-step explanation:
The expression cos(sin-1(6/x)) involves a trigonometric function and its inverse. Since sin-1(y) represents the angle whose sine is y, the expression is asking for the cosine of that angle. To visualize this, consider a right-angled triangle where the opposite side over the hypotenuse equals 6/x.
For such a triangle, we can let the length of the opposite side to be 6 and the hypotenuse to be x. By the Pythagorean theorem, we can find the adjacent side by a2 = x2 - 62. So, the cosine of the angle is the ratio of the adjacent side over the hypotenuse, which will be a/x, where a is the length of the adjacent side calculated using the Pythagorean theorem.
However, it's important to note that the value 6/x must satisfy -1 ≤ sin(θ) ≤ 1 since the sine function only outputs values in this range. Therefore, for the value 6/x to be valid, x must be at least 6, ensuring that sin(sin-1(6/x)) ≤ 1.