Question

By means of a free-hand drawing of the graph estimate the value of each of the following functions. Remember that a curve recedes from its tangent very slowly near a point of inflection. a) \(\sin ^{-1} \cdot 113\) b) \(\tan ^{-1}(-.214)\); c) \(\cos ^{-1} .172\); d) \(\tan ^{-1}(-7.4)\); e) \(\cot ^{-1}(-.152)\); \(f\) ) \(\cos ^{-1}(-.998)\) g) \(\sin ^{-1}(-.21)\); h) \(\sin ^{-1} .89\); i) \(\tan ^{-1} 5.2\); j) \(\cot ^{-1} 7.3 ;\) k) \(\cos ^{-1}(-.138)\); l) \(\sin ^{-1}(-.138)\). In what cases is your error large; in what, small?

Short answer

Answer: The error in estimation will be large when the graph of the inverse trigonometric function is rapidly changing, typically when the angle is around \(0\) or \(\pm \frac{\pi}{2}\) for inverse sin and tan functions, and around \(\frac{\pi}{2}\) for inverse cos and cot functions. The error will likely be smaller when the graph has a more moderate slope, which is typically when the angle is away from \(0\) or \(\pm \frac{\pi}{2}\) for inverse sin and tan functions, and around \(0\) for inverse cos and cot functions.

Step-by-step solution

Identify the function and its inverse

For each function, we need to identify the trigonometric function and its inverse.

Sketch a rough graph of the inverse function

For each inverse trigonometric function, sketch a rough graph of the function. You can use the following information to help you: - Inverse sin and inverse cos functions have domains in the range of \([-1, 1]\) and their ranges are between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) for inverse sin and \([0, \pi]\) for inverse cos. - Inverse tan and cot functions have domains in all real numbers, and their ranges are between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) for inverse tan and \((0, \pi)\) for inverse cot.

Find the value of the function near the given point

Locate the given point on the x-axis of your sketch of the inverse trigonometric function. Observe the behavior of the graph near this point, and find the corresponding y-value.

Estimate the value of the inverse trigonometric function

For each function, write down the estimated value of the inverse function at the given point based on your graph. After finding the approximate values for each of the given functions, we will analyze the errors of the estimations:

Error Analysis

The error in estimation will be large in cases where the graph of the inverse trigonometric function is rapidly changing. This typically occurs when the angle is around \(0\) or \(\pm \frac{\pi}{2}\) for inverse sin, and tan functions, and around \(\frac{\pi}{2}\) for inverse cos and cot functions. Conversely, the error will likely be smaller when the graph has a more moderate slope, which is typically when the angle is away from \(0\) or \(\pm \frac{\pi}{2}\) for inverse sin and tan functions, and around \(0\) for inverse cos and cot functions.

Key concepts

Graphing Inverse Trigonometric Functions

Graphing inverse trigonometric functions is pivotal for understanding the relation between an angle and its trigonometric ratio, particularly when solving equations and estimating values. For instance, Arcsine (sin^-1): This function reflects sin^x over the line y = x, which results in a graph that arcs from -π/2 to π/2 in the y-axis for the given domain of [-1, 1].
Arctangent (tan^-1): By graphing this function, one can observe its approach to -π/2 and π/2 as x moves further away from zero in both directions. The graph stretches infinitely along the x-axis, since tan^x is not bounded.

Graphing these functions can help students with estimation tasks by visualizing the curve's behavior near points of interest, such as points of inflection or steep regions.

Domains of Inverse Trigonometric Functions

Understanding the domains of inverse trigonometric functions is essential for correctly sketching the graphs and finding the values of inverse functions. Because these functions are inverses, their domains are the ranges of the original trigonometric functions. For example, the domain of sin^-1(x) is limited from -1 to 1 since the sine of an angle never exceeds this interval. Similarly, tan^-1(x) and cot^-1(x) can take any real number as input, as the tangent and cotangent functions have outputs that span all real numbers.

When graphing, always consider the domain to avoid plotting undefined values. If you're sketching these on a coordinate grid, mark the domain clearly to represent the function accurately.

Ranges of Inverse Trigonometric Functions

The ranges of inverse trigonometric functions, which are also the domains of the original trigonometric functions, define where the output values will fall. For instance, sin^-1(x) has a range from -π/2 to π/2, meaning that the arcsine of any number in the domain will yield an angle within these bounds. Contrastingly, cos^-1(x) offers an angle between 0 and π, in accordance with the fact that cosine values fall between -1 and 1.

Remembering these ranges helps in finding the expected estimation output when performing free-hand sketches of the graphs during exercises, ensuring more accurate results.

Error Analysis in Function Estimation

Conducting error analysis in estimating values of inverse trigonometric functions involves recognizing where the graph's slope is more dramatic. The steepness affects estimation accuracy since small horizontal changes lead to significant vertical shifts. It is here where the tangent line recedes slowly from the curve, making the error in estimation quite large. On the other hand, in the flatter regions where the curve is less sensitive to small changes in the input value, estimate errors are usually smaller.

Students should pay particular attention when the provided value approaches the limits of the function's domain, as the function's rate of change tends to infinity, causing heightened estimation error. By recognizing these patterns and points of inflection, students can improve their estimation techniques and reduce the potential for error in their work.