Problem 3 - Precalculus Mathematics for Calculus

Question

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} 1\) (b) \(\sin ^{-1} \frac{\sqrt{3}}{2}\) (c) \(\sin ^{-1} 2\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{2}\); (b) \(\frac{\pi}{3}\); (c) undefined.

Step 1: Understanding Inverse Sine Function

The inverse sine function, denoted as \(\sin^{-1}(x)\), gives the angle \(\theta\) whose sine value is \(x\). Its range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This means any output will be in this interval.

Step 2: Solving \(\sin^{-1}(1)\)

We need to find the angle whose sine is 1. The sine of \(\frac{\pi}{2}\) is 1. Therefore, \(\sin^{-1}(1) = \frac{\pi}{2}\).

Step 3: Solving \(\sin^{-1}(\frac{\sqrt{3}}{2})\)

We need to find the angle whose sine is \(\frac{\sqrt{3}}{2}\). This value is commonly recognized from the unit circle. The angle \(\frac{\pi}{3}\) has a sine value of \(\frac{\sqrt{3}}{2}\). Thus, \(\sin^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}\).

Step 4: Checking \(\sin^{-1}(2)\)

The sine function outputs values only in the range \([-1, 1]\). Since 2 is outside this range, \(\sin^{-1}(2)\) is undefined.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental component of trigonometry, mapping any given angle to a specific ratio of two sides of a right-angled triangle.

• The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse.
• It works closely with the other trigonometric functions such as cosine and tangent.

When we introduce the concept of the inverse sine, denoted as \( \sin^{-1}(x) \), we are essentially finding the angle corresponding to a given sine value.The sine function itself is periodic and oscillates between -1 and 1. This oscillation contributes to our understanding that each sine value within this range corresponds to a specific angle on the unit circle.

The Role of the Unit Circle

The unit circle is a critical tool in trigonometry and plays an essential role in understanding trigonometric functions.

• A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.
• Any point on the circle has coordinates \((\cos\theta, \sin\theta)\), where \(\theta\) is the angle formed with the positive x-axis.

The unit circle helps visualize the values and relationships of trigonometric functions, including sine. When evaluating an inverse sine operation like \(\sin^{-1}(x)\), the unit circle aids in identifying which angle corresponds to that sine value through observed coordinates. For instance, the point associated with \(\frac{\sqrt{3}}{2}\) on the unit circle helps determine that the angle is \(\frac{\pi}{3}\) in the context of the sine function.

Exploring the Function Range

The function range of sine is limited to values between -1 and 1, inclusive. This restriction has significant implications when dealing with inverse sine functions.

• The range of \(\sin(x)\) is \([-1, 1]\), meaning the function will never yield a value less than -1 or greater than 1.
• When using inverse sine, only values within this range return an angle as output.

For instance, when we attempt to compute \(\sin^{-1}(2)\), it becomes immediately apparent that this value is undefined. Since 2 falls outside the allowed range, there is no angle for which the sine value equates to 2.

Angle Measurement in Trigonometry

Angles in trigonometry are typically measured in radians or degrees, with radians being the standard unit in many mathematical contexts.

• One full circle rotation equals \(2\pi\) radians or 360 degrees.
• Sine and inverse sine functions in mathematics often yield angles in radians.

For example, finding \(\sin^{-1}(1)\) would give \(\frac{\pi}{2}\) radians as it corresponds to the angle with a sine value of 1 on the unit circle. Understanding radians in this context is crucial, especially when transitioning between angular measurement systems. Leveraging both degrees and radians allows for a comprehensive approach to solving problems involving angles and trigonometric functions.