Question

\(y=3 \csc \left(1-\frac{x}{3}\right), \quad x=1, \quad d x=0.1\)

Short answer

The final calculation of the function is the product of 3 and the reciprocal of the sine of 2/3.

Step-by-step solution

Understand the Function

Firstly, the function provided, \(y = 3 \csc \left(1-\frac{x}{3}\right)\) uses the trigonometric function cosecant, which is the reciprocal of the sine function.

Substitute the values for x and dx

Substitute \(x = 1\) and \(dx = 0.1\) into the function. Therefore, it becomes: \(y = 3 \csc \left(1-\frac{1}{3}\right)\). Simplify the fraction inside the bracket so that it becomes: \(y = 3 \csc \left(\frac{2}{3}\right)\).

Calculate the cosecant of 2/3

Now, calculate the cosecant of \(\frac{2}{3}\). As mentioned before, cosecant is the reciprocal of the sine function. So, we need to calculate \(\csc \left(\frac{2}{3}\right) = 1 / \sin \left(\frac{2}{3}\right)\).

Final Calculation

Now, multiply 3 with the obtained value of \(\csc \left(\frac{2}{3}\right)\) to get the final value of y.