91Ó°ÊÓ

The inverse sine function, often denoted as \( \sin^{-1}(x) \) or arcsin, is used to find the angle whose sine is a given number. This function is particularly useful when solving trigonometric equations. In the exercise, we are asked to solve \( \sin (2\theta) = -0.7843 \). Here, we employ the inverse sine function to isolate \( 2\theta \) by rewriting it as \( 2\theta = \sin^{-1}(-0.7843) \).
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It's important to note that the output of \( \sin^{-1} \) falls within the range of \(-90^{\circ}\) to \(90^{\circ}\), due to its definition on the unit circle. This provides the principal value, or the most direct angle representation of the trigonometric ratio.
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In our case, the calculation gives us \( 2\theta \approx -51.55^{\circ} \). However, since trigonometric functions are periodic, we can generate the required angles by understanding their symmetry and periodic nature.

Sine is a periodic function, meaning it repeats its values in regular intervals. For sine, this interval, or period, is \( 360^{\circ} \). Understanding this period is crucial while solving equations like \( \sin (2\theta) = -0.7843 \) because it helps identify the cycles where these angles recur.
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With the initial solution \( 2\theta \approx -51.55^{\circ} \), we can determine additional solutions by adding or subtracting full periods: \( 360^{\circ} \). This changes the angle but not the sine value.
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For example, adding \( 360^{\circ} \) to \(-51.55^{\circ}\) yields \( 308.45^{\circ} \), bringing the solution within the principal cycle \( 0^{\circ} \leq 2\theta < 360^{\circ} \). This is essential in trigonometry, as recognizing the periodicity allows one to find all possible solutions within a specified range.

The concept of angle symmetry in trigonometry is pivotal, particularly for the sine function. Sine symmetry indicates that the sine of an angle in a circle's quadrant is equivalent to the sine of its supplementary angle. Mathematically, this is conceived as \( \sin(x) = \sin(180^{\circ} - x) \).
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This property was used in the given exercise to find another value for \( 2\theta \). Starting from \( 2\theta = -51.55^{\circ} \), we applied the symmetry property to derive \( 2\theta = 180^{\circ} + 51.55^{\circ} = 231.55^{\circ} \). This additional solution represents the angle's symmetry about the line \( y = \tan(90^{\circ}) \), a characteristic feature of trigonometric functions.
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Understanding angle symmetry not only aids in solving equations like the one given but also deepens comprehension of geometric interpretations across unit circles, which is central in trigonometric studies.