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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 29
Chapter 2, Problem 29

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ (-0.32)

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Identify the function involved: here, \( \sin^{-1} \) represents the inverse sine function, also known as arcsine, which gives the angle whose sine is the given value.
Recall the domain and range of the inverse sine function: the input value must be between -1 and 1, and the output angle will be in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) or equivalently \( [-90^\circ, 90^\circ] \). Since -0.32 is within the domain, the calculation is valid.
Use a calculator set to the correct mode (degrees or radians depending on the problem context) to find \( \sin^{-1}(-0.32) \).
Input the value -0.32 into the inverse sine function on the calculator to get the angle whose sine is -0.32.
Round the resulting angle to two decimal places as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted as sin⁻¹ or arcsin, returns the angle whose sine is a given number. It is defined for inputs between -1 and 1 and outputs angles typically in the range of -90° to 90° (or -π/2 to π/2 radians). Understanding this helps in finding the angle corresponding to a sine value.
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Inverse Sine

Domain and Range of the Inverse Sine Function

The domain of sin⁻¹ is limited to values between -1 and 1 because sine values cannot exceed this range. The range of sin⁻¹ is the set of possible output angles, usually from -90° to 90°. Recognizing these limits ensures the input is valid and the output angle is interpreted correctly.
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Domain and Range of Function Transformations

Using a Calculator to Evaluate Inverse Trigonometric Functions

Calculators have specific modes (degree or radian) that affect the output of inverse trig functions. To find sin⁻¹(-0.32), input the value correctly and ensure the calculator is set to the desired angle unit. Rounding the result to two decimal places provides a precise and usable answer.
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Related Practice
Textbook Question

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = −4 sin 3π/2 t

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Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 3 csc x

Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2πx + 4π)

Textbook Question

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.


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Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 3 sin(πx + 2)

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Textbook Question

In Exercises 29–36, find the length x to the nearest whole unit.

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