Problem 54
Question
Use a calculator to give each real-number value of \(y .\) $$y=\arcsin 0.81926439$$
Step-by-Step Solution
Verified Answer
The angle \( y \) is approximately \( 55.014^{\circ} \).
1Step 1: Understand the problem
We need to find the value of the angle \( y \) whose sine is \( 0.81926439 \). In other words, we are looking for \( y = \arcsin(0.81926439) \). This means finding an angle \( y \) such that \( \sin(y) = 0.81926439 \). Since the \( \arcsin \) function is the inverse of \( \sin \), it gives us the angle for a given sine value.
2Step 2: Use a calculator to find the angle
Turn on your scientific calculator. Ensure it is set to degree mode if you need the angle in degrees. Enter the value \( 0.81926439 \) and then press the \( \arcsin \) (or \( \sin^{-1} \)) function key. The calculator will provide the value of \( y \).
3Step 3: Interpret the result
The calculator computes \( y = \arcsin(0.81926439) \) and provides the angle \( y \). For this value, \( y \approx 55.014^{\circ} \), indicating that this is the angle in degrees whose sine is close to \( 0.81926439 \).
Key Concepts
ArcsinAngle CalculationSine Function
Arcsin
The term "arcsin" stands for the inverse sine function. In trigonometry, the inverse functions are used to find angles when the value of a trigonometric function is known. In particular, the arcsin function, denoted as \( \arcsin(x) \) or sometimes \( \sin^{-1}(x) \), is used to determine the angle whose sine is a specified number.
It's important to remember that sine, like other trigonometric functions, maps an angle to its ratio, which is a value between -1 and 1. Therefore, the arcsin function will only take values between -1 and 1 as input. This means if \( x \) is not within this range, the arcsin is undefined. The output of \( \arcsin(x) \) is the angle \( y \) in the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or \([-90^\circ, 90^\circ]\) when using degrees). This range is chosen because it uniquely represents each possible angle value that sine can achieve within -1 to 1.
When your calculator shows an angle, it is this specific angle within the principal range for which the sine value equals the information given to the arcsin function.
It's important to remember that sine, like other trigonometric functions, maps an angle to its ratio, which is a value between -1 and 1. Therefore, the arcsin function will only take values between -1 and 1 as input. This means if \( x \) is not within this range, the arcsin is undefined. The output of \( \arcsin(x) \) is the angle \( y \) in the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or \([-90^\circ, 90^\circ]\) when using degrees). This range is chosen because it uniquely represents each possible angle value that sine can achieve within -1 to 1.
When your calculator shows an angle, it is this specific angle within the principal range for which the sine value equals the information given to the arcsin function.
Angle Calculation
Calculating an angle involves understanding the relationship between the trigonometric function values and their corresponding angles. In this problem, you need to find the angle \( y \) that makes \( \sin(y) = 0.81926439 \). This is the essence of angle calculation using an inverse function like arcsin.
To perform this calculation, follow these steps:
To perform this calculation, follow these steps:
- Ensure your scientific calculator is in the correct mode, usually angle mode (degrees or radians), that you are interested in. So, if you need the angle in degrees (as is often the case), ensure it's set to degrees.
- Input the sine value, which is 0.81926439 in this instance.
- Press the \( \arcsin \) or \( \sin^{-1} \) button. This calculator function directly provides the angle for that sine value.
Sine Function
The sine function is a fundamental concept in trigonometry that relates a given angle to the ratio of the length of the opposite side to the hypotenuse in a right triangle. Denoted as \( \sin(\theta) \), it is one of the primary trigonometric functions that also include cosine and tangent.
Sine function's importance lies in its periodic properties and its ability to model periodic phenomena in the real world. For an angle \( \theta \), \( \sin(\theta) \) produces values between -1 and 1. This nature makes it a perfect candidate for representing oscillating systems like waves in physics.
When dealing with arcsin or inverse sine, it is important to recognize that sine itself is restricted to this -1 to 1 range, which dictates the valid input for arcsin. Once you input a valid sine value into the function \( \sin^{-1}(x) \), it gives back an angle that, when used in the sine function, returns the original sine value. This inverse relationship is what makes the sine and arcsin functions extremely useful in calculations involving angles and periodic functions.
Sine function's importance lies in its periodic properties and its ability to model periodic phenomena in the real world. For an angle \( \theta \), \( \sin(\theta) \) produces values between -1 and 1. This nature makes it a perfect candidate for representing oscillating systems like waves in physics.
When dealing with arcsin or inverse sine, it is important to recognize that sine itself is restricted to this -1 to 1 range, which dictates the valid input for arcsin. Once you input a valid sine value into the function \( \sin^{-1}(x) \), it gives back an angle that, when used in the sine function, returns the original sine value. This inverse relationship is what makes the sine and arcsin functions extremely useful in calculations involving angles and periodic functions.
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