Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\sin ^{-1}\left(-\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The approximation of the expression \(\sin^{-1}(-3/4)\) rounded to the nearest hundredth is -0.85 in radians or -48.59 in degrees, depending on calculator mode.

Step 1: Substitute the given value in the arcsine function

Substitute the value \(-3/4\) in the arcsine function, which will look like this: \(\sin^{-1}(-3/4)\).

Step 2: Calculate the inverse sine

Using a scientific calculator, calculate the value of \(\sin^{-1}(-3/4)\). This will be your approximated value. Be sure to keep multiple decimal places at first.

Step 3: Round the value

Round the calculated value to the nearest hundredth. This is because the task specifies that rounding should be done to two decimal places.

Step 4: Interpret the result

The result obtained corresponds to an angle in radians if your calculator is set to radian mode, or in degrees if your calculator is set to degree mode. This angle has a sine equal to \(-3/4\).

Key Concepts

Arcsine Function

The arcsine function is one of the inverse trigonometric functions. It is often denoted as \( \sin^{-1}(x) \) or \( \text{asin}(x) \). The arcsine function helps us determine the angle whose sine is a given number. In simpler terms, if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).

• The range of the arcsine function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) radians or \([-90^\circ, 90^\circ]\) degrees.
• It is important to note that the arcsine function is only defined for \( -1 \leq x \leq 1 \), meaning it only takes values within this range as valid inputs.

These limitations ensure that the function has a unique output, which corresponds to an angle for any valid sine value.

Scientific Calculator

Scientific calculators are essential tools for working with trigonometric functions, including inverse trigonometric functions like arcsine. They allow you to perform complex calculations with precision. Here's how to use them effectively for calculating arcsine:

• Start by ensuring your calculator is in the correct mode (radian or degree) based on your needs.
• Locate the arcsine button, often labeled as \( \sin^{-1} \) or \( \text{asin} \). You might need to use a "shift" or "2nd" function to access it.
• Input the value for which you need to find the arcsine, such as \(-\frac{3}{4}\) in our exercise.
• Press the equal button to get the result, which will either be an angle in radians or degrees depending on your calculator mode.

Angle Measurement

Angles can be measured in various units, but the most common are degrees and radians. Understanding these units is crucial when working with trigonometric functions and their inverses.

• Degrees: A circle is divided into 360 degrees. It's a standard unit for measuring angles, especially in everyday contexts.
• Radians: The radian is based on the radius of a circle. There are \(2\pi\) radians in a full circle. Mathematically, radians often simplify equations and formulas.

When working with angles, always pay close attention to which unit you're using. This will influence how you interpret results from trigonometric functions or their inverses.

Radian and Degree Mode

In scientific calculators, you have the option to set the mode to radians or degrees. This setting affects how calculations are interpreted, especially with trigonometric functions.

• Radian Mode: Used mainly in calculus and advanced mathematics. Set your calculator to this mode when your calculations involve radian angular measurements.
• Degree Mode: More intuitive for general angle measurement and used primarily in practical, real-world situations. Use this mode when dealing with problems framed in degrees.

Check the mode on your calculator before performing calculations like \( \sin^{-1}(-\frac{3}{4}) \). Based on your need, decide if you're solving for an angle in radians or degrees. This step is crucial to ensure you interpret your results correctly.