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Chapter 5: Problem 47

Find the exact value of the expression, if it is defined. $$\sin \left(\tan ^{-1}(-1)\right)$$

Short Answer

Expert verified
The exact value is \(-\frac{\sqrt{2}}{2}\).

Step by step solution

01

Identify the Inverse Tangent

The expression involves \( \tan^{-1}(-1) \), which means we need to find the angle \( \theta \) such that \( \tan(\theta) = -1 \). The standard angle for this is \( \theta = -\frac{\pi}{4} \) in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), where the inverse tangent is defined.
02

Find the Sine of the Angle

Now that we know \( \theta = -\frac{\pi}{4} \), we need to find the sine of this angle. \( \sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) \). The sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Thus, \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

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Key Concepts

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

Inverse Trigonometric Functions
Inverse trigonometric functions are fundamental in solving problems involving angles and side lengths in triangles.
  • The inverse tangent function, often written as \( \tan^{-1}(x) \), returns an angle whose tangent is \( x \).
  • This function is typically evaluated within the range \((-\pi/2, \pi/2)\) to ensure each output is unique, making it a "principle value."
  • In simpler terms, when you see \( \tan^{-1}(-1) \), think of the specific angle which, when passed through the tangent function, gives -1.
With this understanding, to solve \( \tan^{-1}(-1) \), you're essentially finding that unique angle, which in this case is \(-\pi/4\), within the specified range.
Angle Properties
Angles have various properties that help us understand their relationships and calculations. When dealing with trigonometric angles:
  • Property of symmetry indicates that trigonometric functions have predictable patterns in different quadrants of a circle.
  • Negative angles, like \(-\frac{\pi}{4}\), imply a clockwise rotation from the positive x-axis. They mirror their positive counterparts but may switch sign depending on the function.
  • For the sine function, \( \sin(-\theta) = -\sin(\theta) \) due to its odd symmetry.
For the expression \( \sin \left( \tan^{-1}(-1) \right) \), knowing these properties simplifies our work. The angle \(-\frac{\pi}{4}\) lies within the negative quadrant, influencing the sign of its sine value.
Unit Circle
The unit circle is a powerful tool to visualize and calculate trigonometric functions. On this circle:
  • The radius is 1, making it easy to define trigonometric ratios based on coordinates.
  • An angle is measured starting from the positive x-axis, rotating counterclockwise for positive angles.
  • The coordinates \((x, y)\) on the unit circle correspond to \((\cos(\theta), \sin(\theta))\) respectively.
Within the unit circle, \(-\frac{\pi}{4}\) lies in the fourth quadrant where cosine is positive and sine is negative. At this angle, both sine and cosine share absolute values as \(\frac{\sqrt{2}}{2}\), given the symmetrical nature of the circle. Thus, for \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), aligning with our previous calculations.

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