Problem 10 - Precalculus Mathematics for Calculus

Question

Find the exact value of each expression, if it is defined. (a) \(\cos ^{-1} 0\) (b) \(\sin ^{-1} 0\) (c) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{2}\); (b) 0; (c) \(-\frac{\pi}{6}\).

Step 1: Understanding Inverse Cosine

The expression \( \cos^{-1} 0 \) asks for the angle whose cosine is 0. The principal value of \( \cos^{-1} x \) is between 0 and \( \pi \) radians. Cosine is 0 at \( \frac{\pi}{2} \) radians.

Step 2: Understanding Inverse Sine for Zero

The expression \( \sin^{-1} 0 \) asks for the angle whose sine is 0. The principal value of \( \sin^{-1} x \) is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians. Sine is 0 at 0 radians.

Step 3: Calculating Inverse Sine of Negative Value

The expression \( \sin^{-1} \left(-\frac{1}{2}\right) \) asks for the angle whose sine is \(-\frac{1}{2}\). The principal value is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). Sine of \(-\frac{1}{2}\) corresponds to \(-\frac{\pi}{6}\) radians.

Key Concepts

Cosine Function and Its Inverse

The cosine function is a fundamental trigonometric function that connects an angle in a right triangle to the ratio of lengths of the adjacent side over the hypotenuse. It's often expressed as \( \cos \theta \). The inverse cosine function, denoted as \( \cos^{-1} x \), allows us to find the angle when we know the cosine's value. It is important to understand that the range of angles for the inverse cosine is restricted to 0 to \( \pi \) radians, ensuring one unique solution for each value.

• The expression \( \cos^{-1}(0) \) is like asking what angle has a cosine of 0. The answer is \( \frac{\pi}{2} \) radians since cosine equals 0 at a right angle in the unit circle.
• The inverse cosine is sometimes called "arc cosine" and is particularly useful when determining angles in modeling periodic phenomena.

Sine Function and Its Inverse

The sine function relates an angle in a right triangle to the ratio of lengths of the opposite side over the hypotenuse. Denoted as \( \sin \theta \), this function is critical for many calculations and applications. The inverse sine, or \( \sin^{-1} x \), helps find the angle when the sine value is known. The range for inverse sine values is limited to \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians.

• In the case of \( \sin^{-1}(0) \), we are looking for an angle where the sine is zero. The answer is 0 radians, found at the origin on the unit circle.
• The inverse sine function is often used in engineering and physics to resolve angle-related problems where only the sine value is provided.

Understanding Angle Measurement with Inverse Trigonometric Functions

Angles can be measured in various units, but radians are the standard in mathematics. One full circle is \( 2\pi \) radians, equivalent to 360 degrees. When dealing with inverse trigonometric functions, angles are generally expressed in radians for consistency.

• For instance, \( \sin^{-1}\left(-\frac{1}{2}\right) \) asks what angle in radians, between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), has a sine of \(-\frac{1}{2}\). The answer here is \(-\frac{\pi}{6}\) radians.
• Understanding angle measurement is crucial for converting between degrees and radians when necessary. Remember, \( \pi \) radians equals 180 degrees.

This appreciation for angle measures allows coherent solutions when working with real-world problems involving trigonometric calculations.