Problem 13 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Inverse Function

Each part involves an expression with an inverse trigonometric function. Remember that for an expression like \(\sin(\cos^{-1}(x))\), it indicates that we're finding the sine of the angle whose cosine is \(x\). Similarly, for \(\cos(\tan^{-1}(x))\), we find the cosine of the angle whose tangent is \(x\).

Step 2: Solving Part (a)

For \(a = \sin\left[\cos^{-1}\left(-\frac{1}{2}\right)\right]\), we can find \(\theta\) such that \(\cos \theta = -\frac{1}{2}\). The angle \(\theta\) lies in the second quadrant since cosine is negative. Recall \(\cos \theta = -\frac{1}{2}\) at \(\theta = \frac{2\pi}{3}\). Now find \(\sin \left(\frac{2\pi}{3}\right)\) = \(\sin(\pi - \frac{\pi}{3}) = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\).

Step 3: Solving Part (b)

For \(b = \cos\left(\tan^{-1} 1\right)\), find \(\theta\) such that \(\tan \theta = 1\). This occurs when \(\theta \) is \(\frac{\pi}{4}\). Then, \(\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 4: Solving Part (c)

For \(c = \tan\left[\sin^{-1}(-1)\right]\), find \(\theta\) such that \(\sin \theta = -1\). This angle is \(\theta = -\frac{\pi}{2}\). Then, \(\tan(-\frac{\pi}{2})\) is undefined because the angle represents vertical asymptotes on the tangent curve.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. These identities are vital tools in simplifying trigonometric expressions and in solving complex trigonometric equations.

Some of the most commonly used trigonometric identities include:

• The Pythagorean Identity: \ \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal Identities: \ \( \sin \theta = \frac{1}{\csc \theta} \), \( \cos \theta = \frac{1}{\sec \theta} \), \( \tan \theta = \frac{1}{\cot \theta} \).
• Quotient Identities: \ \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These identities help in understanding complex trigonometric expressions, like those involving inverse functions. Remember, when dealing with inverse trigonometric functions such as \( \sin(\cos^{-1}(x)) \) or \( \cos(\tan^{-1}(x)) \), you are essentially finding the sine or cosine of an angle derived from another trigonometric measurement.

Sine and Cosine Functions

The sine and cosine functions are two of the fundamental trigonometric functions, often employed to describe the relationship between the angles and sides of a right triangle. In the unit circle, the sine of an angle gives the y-coordinate, while the cosine provides the x-coordinate of the point on the circle.

For any angle \( \theta \):

• \( \sin \theta \) measures the vertical component, interpreted as the length of the opposite side over the hypotenuse in a right triangle.
• \( \cos \theta \) measures the horizontal component, interpreted as the length of the adjacent side over the hypotenuse.

When we consider inverse trigonometric functions like \( \sin(\cos^{-1}(-\frac{1}{2})) \), we are determining the sine of the angle \( \theta \) where the cosine is \(-\frac{1}{2}\). We often need to recognize certain key angle values:

• \( \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)
• \( \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)

These values are useful when evaluating expressions that involve inverse functions.

Angles in Quadrants

Understanding the angles and their respective functions within different quadrants is crucial when working with inverse trigonometric functions. The coordinate plane is divided into four quadrants, each with different sign conventions for the trigonometric functions:

• Quadrant I: Both \( \sin \theta \) and \( \cos \theta \) are positive.
• Quadrant II: \( \sin \theta \) is positive, while \( \cos \theta \) is negative.
• Quadrant III: Both \( \sin \theta \) and \( \cos \theta \) are negative.
• Quadrant IV: \( \sin \theta \) is negative, while \( \cos \theta \) is positive.

In solving problems like \( \sin(\cos^{-1}(-\frac{1}{2})) \), the angle \( \theta \) is determined by the inverse cosine in the second quadrant, since cosine is negative. Knowing that cosine's negativity places \( \theta \) in Quadrant II helps find that \( \theta = \frac{2\pi}{3} \), where the sine function is indeed positive as expected by quadrant signs. Thus, knowledge of angles in quadrants aids in accurately interpreting inverse trigonometric expressions.