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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.6.3c
Capitolo 7, Problema 7.6.3c

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. c. sin^(-1)(-√3/2)

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Recognize that the problem asks for the angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \). This means we are looking for the inverse sine (arcsin) of \( -\frac{\sqrt{3}}{2} \).
Recall that the sine function is negative in the third and fourth quadrants. Since the range of \( \sin^{-1} \) (arcsin) is restricted to \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), the angle must lie in either the fourth quadrant (negative angles) or the first quadrant (positive angles, but sine positive there). So here, the angle will be in the fourth quadrant (between \( -\frac{\pi}{2} \) and 0).
Identify the reference angle whose sine is \( \frac{\sqrt{3}}{2} \). From common special angles, \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \). So the reference angle is \( \frac{\pi}{3} \).
Since the sine value is negative and the angle must be in the fourth quadrant (due to the range of arcsin), the angle is \( -\frac{\pi}{3} \). This is the angle whose sine is \( -\frac{\sqrt{3}}{2} \) within the principal range of arcsin.
Express the final answer as \( \theta = \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \).

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Concetti chiave

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Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function returns the angle whose sine value is a given number. Its output is restricted to the range [-π/2, π/2] (or [-90°, 90°]) to ensure it is a function. Understanding this helps find the principal angle corresponding to a sine value.
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Inverse Sine

Reference Triangles and Quadrants

Reference triangles are right triangles used to find angle measures based on trigonometric ratios. Knowing the quadrant where the angle lies is essential because sine values are positive or negative depending on the quadrant, affecting the angle's actual measure.
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Trig Values in Quadrants II, III, & IV

Sign of Sine in Different Quadrants

Sine is positive in the first and second quadrants and negative in the third and fourth quadrants. Since sin⁻¹(-√3/2) is negative, the angle must lie in either the fourth or first quadrant (considering the principal range), guiding the correct angle determination.
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Inverse Sine
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86. This exercise explores the difference between

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7. c. arcsec(-2)

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c. 1/√x

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80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

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