Problem 43 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(-\frac{\sqrt{2}}{2}\)

Step 1: Understand the Problem

We need to find the exact value of \( \sin(\tan^{-1}(-1)) \). This involves determining the sine of an angle whose tangent is -1.

Step 2: Determine the Angle from Inverse Tangent

The expression \( \tan^{-1}(-1) \) returns the angle whose tangent value is -1. In trigonometric terms, this corresponds to an angle of \( -\frac{\pi}{4} \) radians in the interval \(( -\frac{\pi}{2}, \frac{\pi}{2})\). This interval is the range of the inverse tangent function.

Step 3: Calculate the Sine of the Angle

We need to find \( \sin \left( -\frac{\pi}{4} \right) \). Using the unit circle or basic trigonometric identities, \( \sin \left( -\frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2} \), because \( \sin(-\theta) = -\sin(\theta) \) and \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential to find angles when the trigonometric ratio is known. They reverse the function process of basic trigonometric functions like sine, cosine, and tangent. For example, the inverse tangent function, denoted as \( \tan^{-1}(x) \), returns an angle whose tangent is \( x \). These functions have specific ranges to ensure each input corresponds to a unique angle output:

• \( \sin^{-1}(x) \) has a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
• \( \cos^{-1}(x) \) has a range of \( [0, \pi] \)
• \( \tan^{-1}(x) \) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)

Understanding inverse trigonometric functions helps in determining the original angle that produces a given trigonometric value, which is vital for solving many geometric and trigonometric problems.

Unit Circle

The unit circle is a circle centered at the origin on the coordinate plane with a radius of one. It's a crucial tool in trigonometry because it helps to define trigonometric functions, even for angles larger than 90 degrees or negative angles.Each point on the unit circle corresponds to an angle \( \theta \), and its coordinates \((\cos(\theta), \sin(\theta))\) represent the cosine and sine of that angle. This representation makes it easy to visualize how these functions behave across different quadrants:

• In the first quadrant, both sine and cosine are positive.
• In the second quadrant, sine is positive, and cosine is negative.
• In the third quadrant, both sine and cosine are negative.
• In the fourth quadrant, sine is negative, and cosine is positive.

Using the unit circle simplifies the calculation of trigonometric values because the hypotenuse is always 1, aligning the circle’s circumference with trigonometric values.

Trigonometry Identities

Trigonometric identities are equations that hold true for any value of the involved variables and serve as useful tools for simplifying trigonometric expressions or solving equations. These identities encompass relationships between trigonometric functions.Here are some fundamental identities:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identities:
  • \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)
  • \( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \)
• Double Angle Identities:
  • \( \sin(2a) = 2\sin(a)\cos(a) \)
  • \( \cos(2a) = \cos^2(a) - \sin^2(a) \)

These identities are foundational to trigonometry and frequently used in algebraic manipulation, allowing us to express trigonometric expressions in various forms to better solve equations or verify complex relationships.