Problem 8 - Precalculus: Functions and Graphs

Question

Find the exact value of the expression whenever It is defined. (a) \(\sin \left(\sin ^{-1} \frac{2}{3}\right)\) (b) \(\cos \left[\cos ^{-1}\left(-\frac{1}{5}\right)\right]\) (c) \(\tan \left[\tan ^{-1}(-9)\right]\)

Step-by-Step Solution

Answer

(a) \(\frac{2}{3}\), (b) \(-\frac{1}{5}\), (c) \(-9\).

Step 1: Identify the Problem

We need to evaluate the expressions for (a) \(\sin(\sin^{-1}(\frac{2}{3}))\)(b) \(\cos(\cos^{-1}(-\frac{1}{5}))\) (c) \(\tan(\tan^{-1}(-9))\)The expressions use inverse trigonometric functions followed by the same trigonometric functions. The task is to find the simplification for each.

Step 2: Evaluate Part (a)

For \(\sin(\sin^{-1}(\frac{2}{3}))\), the expression is an angle whose sine value is \(\frac{2}{3}\). Recall that \(\sin^{-1}\) outputs an angle whose sine value is the input. Hence, \(\sin(\theta) = \frac{2}{3}\). Thus: \[ \sin(\sin^{-1}(\frac{2}{3})) = \frac{2}{3} \]

Step 3: Evaluate Part (b)

For \(\cos(\cos^{-1}(-\frac{1}{5}))\), the expression represents an angle whose cosine value is \(-\frac{1}{5}\). Therefore: \[ \cos(\cos^{-1}(-\frac{1}{5})) = -\frac{1}{5} \] This result comes from the property of inverse functions where applying the function to its inverse gives the original value, as long as it's within the function's range.

Step 4: Evaluate Part (c)

For \(\tan(\tan^{-1}(-9))\), the expression is simplified by knowing that \(\tan^{-1}(-9)\) gives an angle whose tangent is \(-9\), so:\[ \tan(\tan^{-1}(-9)) = -9 \] Here, \(\tan^{-1}\) outputs an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), holding the tangent value as the input.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are useful tools for simplifying and solving equations involving angles and lengths. Inverse trigonometric functions, like \( \sin^{-1}, \cos^{-1}, \tan^{-1} \), are particularly interesting as they allow you to work backwards from a known ratio to find the angle.
When both a trigonometric function and its inverse function are applied successively (e.g., \( \sin(\sin^{-1}(x)) \)), the result is typically \( x \) within the function's domain.

• For example, \( \sin(\sin^{-1}(x)) = x \) if \( -1 \leq x \leq 1 \)
• \( \cos(\cos^{-1}(x)) = x \) if \( -1 \leq x \leq 1 \)
• \( \tan(\tan^{-1}(x)) = x \) for all real \( x \)

These identities are essential for solving problems more efficiently and are particularly useful when dealing with expressions involving both trigonometric functions and their inverses.

Sine Function

The sine function is one of the primary trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. It is often written as \( \sin(\theta) \), where \( \theta \) is the angle.
The inverse operation, \( \sin^{-1} \), is used to find the angle given the sine value, producing an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This can be useful in determining precise angles in various contexts such as physics and engineering.

• By the property of inverse functions:
  \( \sin(\sin^{-1}(x)) = x \), which translates directly into the solution like the situation in part (a) of the exercise.

Remember, this identity only holds when the value of \( x \) is between \(-1 \) and \( 1 \), the range of values \( \sin(x) \) can take.

Cosine Function

The cosine function, closely related to the sine function, deals with the ratio of the adjacent side over the hypotenuse in a right triangle. Notably, \( \cos(\theta) \) holds values between \( -1 \) and \( 1 \). Its inverse, \( \cos^{-1} \), outputs angles within \([0, \pi] \).
In the scenario of the problem with domain constraints:

• Considering \( \cos(\cos^{-1}(x)) = x \) leads us directly to solve expressions like part (b), providing the cosine value of \(-\frac{1}{5} \).

Ensuring that this function works correctly requires the input or output of \( \cos^{-1} \) to lie within its defined range.

Tangent Function

Among the trigonometric functions, the tangent function is defined as the ratio of the sine and cosine functions, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). It can take any real number as a value, making it unique compared to sine and cosine.
The inverse tangent function, \( \tan^{-1} \), yields angles from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This makes computing inverse tangents straightforward for values such as \(-9 \) in this exercise:

• The property \( \tan(\tan^{-1}(x)) = x \) holds true for all real numbers \( x \). This principle simplifies part (c) by recognizing \(-9 \) as the result of the tangent operation's reversal.

Thanks to this property, tangents and their inverses serve as powerful tools for quick conversion between angles and ratios.