Problem 14 - Precalculus: Functions and Graphs

Question

Find the exact value of the expression whenever It is defined. (a) \(\sin \left(\tan ^{-1} \sqrt{3}\right)\) (b) \(\cos \left(\sin ^{-1} 1\right)\) (c) \(\tan \left(\cos ^{-1} 0\right)\)

Step-by-Step Solution

Answer

(a) \( \frac{\sqrt{3}}{2} \), (b) 0, (c) undefined.

Step 1: Understand the Inverse Function

To find the exact values of these expressions, we need to use the properties of inverse trigonometric functions. Each expression involves substituting an inverse trigonometric value into a trigonometric function.

Step 2(a): Simplify \( \sin( \tan^{-1} \sqrt{3}) \)

We begin by setting \( \theta = \tan^{-1} \sqrt{3} \). This means \( \tan \theta = \sqrt{3} \). This matches the known value \( \tan(\frac{\pi}{3}) = \sqrt{3} \), indicating \( \theta = \frac{\pi}{3} \). Use the identity \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \). Thus, \( \sin(\tan^{-1} \sqrt{3}) = \frac{\sqrt{3}}{2} \).

Step 2(b): Simplify \( \cos(\sin^{-1} 1) \)

Set \( \theta = \sin^{-1} 1 \). This implies \( \sin \theta = 1 \), which only happens when \( \theta = \frac{\pi}{2} \). Hence, \( \cos(\sin^{-1} 1) = \cos(\frac{\pi}{2}) = 0 \).

Step 2(c): Simplify \( \tan(\cos^{-1} 0) \)

Set \( \theta = \cos^{-1} 0 \). This implies \( \cos \theta = 0 \), occurring when \( \theta = \frac{\pi}{2} \) or \( \theta = \frac{3\pi}{2} \). Since \( \cos^{-1} \) is confined to \([0, \pi]\), \( \theta = \frac{\pi}{2} \). Therefore, \( \tan(\cos^{-1} 0) = \tan(\frac{\pi}{2}) \), which is undefined as it involves division by zero.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables for which both sides of the equation are defined. These identities are extremely useful when simplifying expressions and proving the equality of two sides of an equation.
Some of the fundamental trigonometric identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Quotient identities like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Reciprocal identities, exemplified by \( \csc \theta = \frac{1}{\sin \theta} \).

These identities help convert one trigonometric function into another, making it easier to work with complex expressions. For example, the identity \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \) is specifically used in cases where a trigonometric function is applied to an inverse trigonometric function, simplifying the computation.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions involve finding the precise trigonometric ratios for specific angles. Instead of using a calculator for approximate values, we rely on known values and identities to determine the exact ratios.
Some common angles with known exact trigonometric values are \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \, \text{and others} \). For example:

• \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \)
• \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)
• \( \tan(\frac{\pi}{3}) = \sqrt{3} \)

By understanding these values, solving expressions such as \( \sin(\tan^{-1}\sqrt{3}) \) becomes straightforward since we know \( \tan(\frac{\pi}{3}) = \sqrt{3} \) and consequently \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).
Grasping these exact values is crucial for students to tackle problems without relying on numerical approximation software.

Properties of Inverse Functions

Inverse functions essentially reverse the action of the original function. For trigonometric functions, the inverse functions are \( \sin^{-1}, \cos^{-1}, \tan^{-1} \), and they return an angle whose trigonometric function equals the given value.
These inverse trigonometric functions have specific ranges to ensure they are functions, meaning each input results in one output.

• The range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• For \( \cos^{-1} \) it is \([0, \pi]\).
• For \( \tan^{-1} \), it's \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Understanding these properties helps in recognizing why certain expressions yield specific results. For instance, \( \tan(\cos^{-1} 0) \) involves finding an angle whose cosine is zero within the range of \( \cos^{-1} \), leading to \( \theta = \frac{\pi}{2} \) which results in an undefined tangent because \( \tan(\frac{\pi}{2}) \) is not defined.