Final answer:
The value of sin x from 0 to 3π/2 is 1.
Explanation:
To evaluate sin x from 0 to 3π/2, we first need to understand the concept of trigonometric functions and their values in different quadrants. In the first quadrant (0 to π/2), all trigonometric functions have positive values. In the second quadrant (π/2 to π), only sine and cosecant have positive values. In the third quadrant (π to 3π/2), only tangent and cotangent have positive values. And in the fourth quadrant (3π/2 to 2π), only cosine and secant have positive values.
Now, in the given problem, we are dealing with the interval from 0 to 3π/2, which falls in the first and second quadrants. Since sine has a positive value in both these quadrants, we can directly conclude that the final answer will be positive.
To further confirm our answer, we can use the unit circle and the trigonometric ratios to calculate the exact value of sin x in this interval. In the first quadrant, the value of sin x is equal to the y-coordinate of the point on the unit circle. Since the point on the unit circle at 0 degrees has a y-coordinate of 0, we can say that sin 0 = 0.
Moving on to the second quadrant, we need to find the value of sin x at π/2. Again, using the unit circle, we can see that the y-coordinate of the point at π/2 is 1. Therefore, sin π/2 = 1.
In the interval from 0 to 3π/2, we can see that the value of x increases from 0 to π/2 and then decreases back to 0. This means that the value of sin x will also increase and then decrease back to 0. At π, the value of sin x will be 0 again as the y-coordinate of the point at π is 0. And at 3π/2, the value of sin x will be -1 as the y-coordinate of the point at 3π/2 is -1.
Hence, we can see that as x increases from 0 to 3π/2, the value of sin x increases from 0 to 1 and then decreases back to 0. Therefore, the final answer for the given problem is 1.