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Chapter 8: Problem 9

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) \(\sin ^{-1}(0.13844)\) (b) \(\cos ^{-1}(-0.92761)\)

Short Answer

Expert verified
(a) \( \sin^{-1}(0.13844) \approx 0.13879 \) radians; (b) \( \cos^{-1}(-0.92761) \approx 2.76109 \) radians.

Step by step solution

01

Understanding Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1}(x) \), is used to find the angle whose sine value is \( x \). The range of \( \sin^{-1}(x) \) is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians or \(-90\) to \(90\) degrees.
02

Compute \( \sin^{-1}(0.13844) \) Using Calculator

To find \( \sin^{-1}(0.13844) \), enter the value \( 0.13844 \) into the calculator and use the inverse sine function (usually labelled as \( \sin^{-1} \) or \( \arcsin \)). The calculator provides the output in radians unless set to degrees.
03

Round the Value to Five Decimal Places

Once you obtain \( \sin^{-1}(0.13844) \), round the result to five decimal places. The approximate value in radians is 0.13879.
04

Understanding Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1}(x) \), finds the angle whose cosine is \( x \). The range of \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \) radians or \(0\) to \(180\) degrees.
05

Compute \( \cos^{-1}(-0.92761) \) Using Calculator

To find \( \cos^{-1}(-0.92761) \), enter the value \( -0.92761 \) into the calculator and use the inverse cosine function (typically labeled as \( \cos^{-1} \) or \( \arccos \)). The answer will be provided in radians.
06

Round the Value to Five Decimal Places

After computing \( \cos^{-1}(-0.92761) \), round the result to five decimal places. The approximate value in radians is 2.76109.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Sine Function
The inverse sine function, often represented as \( \sin^{-1}(x) \) or "arcsine," helps us determine the angle whose sine equals \( x \). This function is essential for solving equations where the sine value is known, but the angle is not.
  • It is crucial to understand its limited range, which spans from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) in radian measure. This converts to angles between \(-90\) and \(90\) degrees.
  • When dealing with the inverse sine function on a calculator, ensure your device is set to the desired unit (radians or degrees), as the outputs depend on this.
  • In terms of usage, type the known value (like \(0.13844\)) into the calculator and apply \( \sin^{-1} \) to find the corresponding angle.
  • Remember to round the results to the required decimal places for precision, especially in academic exercises.
Inverse Cosine Function
The inverse cosine function, denoted as \( \cos^{-1}(x) \) or "arccosine," serves to find the angle for a known cosine value. This is particularly useful in trigonometric equations where the angle in question is not immediately known.
  • It has a natural range from \( 0 \) to \( \pi \) radians, which translates to the angles ranging from \(0\) to \(180\) degrees.
  • To effectively use a calculator for this function, first ensure it is in the correct mode (degrees or radians), based on your need. This ensures the results are in the desired format.
  • Insert the cosine value, such as \(-0.92761\), into your calculator and apply the \( \cos^{-1} \) function.
  • Accuracy is key, so make sure to round the result appropriately to five decimal places if specified by the problem.
Radian Measure
Radians are a way to measure angles, fundamentally connected to the unit circle. They offer a more natural way in mathematics to describe angles compared to degrees.
  • In mathematics, the circumference of a circle is \(2\pi\), meaning the entire circle is \(2\pi\) radians.
  • This implies that \( \pi \) radians correspond to \(180\) degrees, establishing the crucial conversion factor: \(1\) radian equals approximately \(57.2958\) degrees.
  • Many mathematical calculations, particularly in calculus and trigonometry, prefer radians due to their seamless integration with the circle's properties.
  • Understanding how to convert between radians and degrees is vital, as it ensures precise and accurate results when working with trigonometric and geometric functions.
By grasping the concept of radians, you can better understand angles and measurements intrinsic to dynamic functions like the inverse sine and cosine functions.

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