Final answer:
To determine the value of 1 + 3x + 9x² + 27x³ +81x⁴ + 243x⁵, where x = cotθ and sinθ = 3/5, we use trigonometric identities to find x = 4/3, then substitute and simplify the expression resulting in a value of 364.
Step-by-step explanation:
If we have 0 < θ < 90°, sinθ = 3/5, and x = cotθ, then to find the value of 1 + 3x + 9x² + 27x³ +81x⁴ + 243x⁵, we can use trigonometric identities and the Pythagorean theorem. Since sinθ = 3/5 for a right-angled triangle, we can deduce that the opposite side is 3, the hypotenuse is 5, and thus, the adjacent side (using the Pythagorean theorem) must be 4. This gives us cosθ = 4/5. Using the definition of cotangent, which is the reciprocal of tangent, or cotθ = cosθ/sinθ, we get x = 4/3.
Now, we substitute x with 4/3 in the given expression: 1 + 3(4/3) + 9(4/3)² + 27(4/3)³ + 81(4/3)⁴ + 243(4/3)⁵. Then we simplify to find the value of this expression. After computing, the final result is that the expression's value is 364.