Problem 12 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\sin ^{-1}\left(-\frac{8}{9}\right)\)

Step-by-Step Solution

Answer

The value of \( \sin^{-1}\left(-\frac{8}{9}\right) \) is approximately \(-1.11977\).

Step 1: Understand the Expression

The expression \( \sin^{-1}\left(-\frac{8}{9}\right) \) is the inverse sine function, also known as arcsin. It gives the angle whose sine value is \(-\frac{8}{9}\). The expression is defined for inputs \( -1 \leq x \leq 1 \).

Step 2: Confirm the Input is within the Domain

Check that \(-\frac{8}{9}\) falls within the domain of the inverse sine function. Since \(-\frac{8}{9} \approx -0.8889\), it is within the range \([-1, 1]\). Thus, it is a valid input for \( \sin^{-1} \).

Step 3: Use a Calculator to Compute the Value

Using a scientific calculator, input \( \sin^{-1}\left(-\frac{8}{9}\right) \) to obtain the approximate angle value. Ensure the calculator is set to radians (unless the question specifies degrees).

Step 4: Round to Five Decimal Places

After calculating, round the result to five decimal places as requested. The approximate value of \( \sin^{-1}\left(-\frac{8}{9}\right) \) is \(-1.11977\).

Key Concepts

Arcsin

Arcsin, or inverse sine function, is a function which helps us find the angle whose sine is a given number. It's particularly useful when you need to work backwards from the sine value to determine the angle itself.

• The notation for arcsin is typically written as \( \sin^{-1}(x) \).
• It is defined only for values \(-1 \leq x \leq 1\), since sine values only fall within this range.
• For example, if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).

Arcsin is crucial in various scientific and engineering applications, especially when interpreting angles in trigonometry. Given \( \sin^{-1}\left(-\frac{8}{9}\right) \), we can determine that the angle which results in a sine of \(-\frac{8}{9}\) is approximately \(-1.11977\) radians.

Sine Function

The sine function is one of the fundamental trigonometric functions, usually denoted as \( \sin(\theta) \). This function relates an angle to the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• The range of the sine function is \([-1, 1]\), meaning it will never output a value outside of this range.
• Sine is periodic with a period of \(2\pi\), so \( \sin(\theta) = \sin(\theta + 2\pi) \).
• In a unit circle, the sine of an angle corresponds to the y-coordinate of a point on the circle.

Understanding the sine function is pivotal for solving trigonometric equations and in analyzing periodic phenomena in physics and engineering.

Trigonometric Calculator

A trigonometric calculator is a handy tool for finding values of trigonometric functions, including their inverses such as arcsin. These calculators not only allow for simple arithmetic but also support trigonometric functions.

• Ensure the calculator is in the correct mode, usually radians or degrees, as specified by the context of the problem.
• Most scientific calculators have a button labeled \( \sin^{-1} \) to find inverse sine values easily.
• They are crucial for verifying manual calculations and ensuring accuracy in trigonometric problems.

In solving the exercise \( \sin^{-1}\left(-\frac{8}{9}\right) \), a trigonometric calculator allows us to quickly find the approximate angle \(-1.11977\), making it an indispensable tool in mathematics education.

Radian Measurement

Radian measurement is a method of measuring angles based on the arc length of a circle. It is widely used in mathematics and physics because it provides a natural unit of measurement when analyzing circular motion and periodic functions.

• One complete revolution around a circle equals \(2\pi\) radians, equivalent to 360 degrees.
• An angle in radians is the length of the arc divided by the radius of the circle.
• Radians provide a direct way to relate linear and angular measures in calculus and trigonometry.

When using trigonometric functions like arcsin, ensuring calculations are in radian mode can provide consistency in mathematical applications. In our problem, \(-1.11977\) radians is a measure of the angle whose sine is \(-\frac{8}{9}\), illustrating the utility of radians in portraying angles.