Evaluate arcsin(8.3/16)

Solution:

Step 1:

Compute the quotient of $8.3$ and $16$. We get $arcsin(0.51875)$.

Step 2:

Find the value of $arcsin(0.51875)$. The result is approximately $0.54538818$ radians.

Knowledge Notes:

To solve the problem of evaluating $\arcsin\left(\frac{8.3}{16}\right)$, we need to understand the following concepts:

1. Inverse Trigonometric Functions: The inverse sine function, denoted as $\arcsin$, is the inverse of the sine function. It takes a value between -1 and 1 and returns an angle whose sine is the given value. The range of $\arcsin$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$ or $[-90^\circ, 90^\circ]$.

2. Radian and Degree: Trigonometric functions can be evaluated in radians or degrees. In this case, the result is given in radians. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. There are $2\pi$ radians in a full circle, which is equivalent to $360^\circ$.

3. Calculating Quotients: The first step in evaluating $\arcsin\left(\frac{8.3}{16}\right)$ is to perform the division of $8.3$ by $16$, which gives us a decimal number. This is the input for the $\arcsin$ function.

4. Using a Calculator: To find the value of $\arcsin(0.51875)$, a scientific calculator or computational software that has the inverse trigonometric functions can be used. The value must be within the domain of the $\arcsin$ function, which is between -1 and 1.

5. Approximation: The value obtained from the calculator is an approximation to the nearest decimal places as specified or allowed by the calculator's precision.

6. Notation: In mathematics, it is important to use parentheses to clearly indicate the argument of a function. For example, $\arcsin(0.51875)$ means we are taking the inverse sine of $0.51875$.

By understanding these concepts, we can evaluate the inverse sine of a given ratio and express the angle in radians.