Problem 5 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{2} \), (b) 0, (c) \pi \).

Step 1: Understand the Problem

The problem asks us to find the exact values of expressions involving inverse trigonometric functions. We need to solve for (a) \( \sin^{-1} 1 \), (b) \( \cos^{-1} 1 \), and (c) \( \cos^{-1}(-1) \). These expressions ask for the angle corresponding to the given trigonometric value.

Step 2: Solve Part (a)

For \( \sin^{-1} 1 \), we want to find an angle \( \theta \) such that \( \sin \theta = 1 \). The sine function equals 1 at \( \frac{\pi}{2} \) in radians (or 90 degrees). Thus, \( \sin^{-1} 1 = \frac{\pi}{2} \).

Step 3: Solve Part (b)

For \( \cos^{-1} 1 \), we need an angle \( \theta \) where \( \cos \theta = 1 \). This occurs at \( \theta = 0 \) radians (or 0 degrees), so \( \cos^{-1} 1 = 0 \).

Step 4: Solve Part (c)

For \( \cos^{-1}(-1) \), we seek an angle \( \theta \) where \( \cos \theta = -1 \). This is true when \( \theta = \pi \) radians (or 180 degrees). Therefore, \( \cos^{-1}(-1) = \pi \).

Step 5: Final Verification and Solutions Summary

Now that we have found each angle, verify the results by considering the range of inverse trigonometric functions: \( \sin^{-1} x \) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), and \( \cos^{-1} x \) ranges from 0 to \(\pi\). Our answers fit these criteria. Thus the solutions are correct.

Key Concepts

Trigonometrical Identities

Trigonometric identities are equations involving trigonometric functions which are true for every value of the occurring variables where both sides of the equality are defined. Understanding these identities is crucial when working with inverse trigonometric functions. Here are some essential points:

• Pythagorean Identity: This is a fundamental identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). It helps to establish relationships between sine and cosine functions.


• Reciprocal Identities: These are formulas like \( \sin \theta = \frac{1}{\csc \theta} \) and \( \cos \theta = \frac{1}{\sec \theta} \), which relate primary trigonometric functions with their reciprocals.


• Inverse Trigonometric Identities: These provide the range and domain for inverse functions. For example, \( \sin^{-1} x \) gives an angle \( \theta \) such that \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \), meaning the output is always an angle in this range.

These identities are vital when solving problems involving inverse trigonometric functions, helping to identify and confirm the solutions.

Radians and Degrees

Radians and degrees are two units of measuring angles. A complete circle is divided into 360 degrees or \( 2\pi \, \text{radians} \). Understanding this conversion is important when working with trigonometric expressions.

• Degrees: They are more intuitive and commonly used in everyday contexts. For instance, a straight angle is 180°.


• Radians: Used mostly in calculus and higher mathematics, radians have the advantage of simplifying many mathematical expressions. For example, one radian is the angle created when the arc length is equal to the radius of the circle.


• Conversion: To convert from degrees to radians, use the formula \( \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \). Conversely, to go from radians to degrees, apply \( \text{Degrees} = \frac{180}{\pi} \times \text{Radians} \).

In the context of inverse trigonometric functions, recognizing the ranges in radians is key, as these often use radians rather than degrees, especially in mathematical solutions.

Exact Trigonometric Values

Exact trigonometric values are often used in problems involving inverse trigonometric functions. These are values for sine, cosine, or tangent that produce simple and exact results, typically associated with well-known angles.

• Sine and Cosine Values: For angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \) (or 0°, 30°, 45°, 60°, 90°), these functions yield precise values such as 0, 1/2, \( \frac{\sqrt{2}}{2} \), \( \frac{\sqrt{3}}{2} \), and 1.


• Inverse Functions: Determining \( \sin^{-1} 1 \) suggests finding an angle where the sine is 1, clearly \( \frac{\pi}{2} \). Similarly, finding \( \cos^{-1} 1 \) or \( \cos^{-1}(-1) \) means locating angles where cosine equals these values, yielding specific results like 0 or \( \pi \).


• Application: Using exact trigonometric values ensures accuracy in solutions without approximations. In exercises like the original problem, knowing these values simplifies finding inverse results.

These exact values are an essential tool for anyone studying trigonometry as they allow solving inverse trigonometric problems with precision.