Problem 70 - Algebra 2

Question

a. Critical Thinking Which expression gives the correct value of \(\csc 60^{\circ} ?\) I. \(\sin \left(\left(60^{-1}\right)^{\circ}\right)\) II. \(\left(\sin 60^{\circ}\right)^{-1}\) III. \(\left(\cos 60^{\circ}\right)^{-1}\) b. Which expression in part (a) represents \(\sin \left(\frac{1}{60}\right) ?\)

Step-by-Step Solution

Answer

a. The correct expression for \(\csc 60^{\circ}\) is \(\left(\sin 60^{\circ}\right)^{-1}\)\n b. The expression \(\sin \left(\left(60^{-1}\right)^{\circ}\right)\) represents \(\sin \left(\frac{1}{60}\right).\)

Step 1: Analyze the cosecant function

The cosecant function, \(\csc\), is the reciprocal of the sine function. That is, \(\csc \theta = \frac{1}{\sin \theta}\), where \(\theta\) is the angle in degrees or radians which is given as \(60^{\circ}\). Thus, the correct expression for \(\csc 60^{\circ}\) is \(\left(\sin 60^{\circ}\right)^{-1}\). This rules out option I and option III.

Step 2: Evaluate the sine function

The sine function, \(\sin\), is positive in the first quadrant, which includes \(60^{\circ}\). Thus, \(\sin 60^{\circ}\) is positive while \(\sin \left(\frac{1}{60}\right)\) is the sine function of an angle converted from degrees to radians and is still a small positive number. Comparing this to the expressions in part (a), the expression that matches is \(\sin \left(\left(60^{-1}\right)^{\circ}\right)\). This is equivalent to option I.

Key Concepts

Cosecant Function

The cosecant function is one of the six fundamental trigonometric functions. It is not as commonly used as sine or cosine but can be extremely important in certain calculations. The function is symbolized as \( \csc \theta \), where \( \theta \) is the angle. Simply put, the cosecant of an angle is the reciprocal of the sine of that angle. Therefore, it is defined by the formula:

• \( \csc \theta = \frac{1}{\sin \theta} \)

The cosecant function only exists when the sine function is non-zero, as division by zero is undefined. It is essential to understand this reciprocal relationship because it helps in simplifying expressions and solving trigonometric equations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include the cosecant, secant, and cotangent. They are directly derived from sine, cosine, and tangent, respectively. Here’s a quick look:

• Cosecant (\( \csc \theta \)): \( \csc \theta = \frac{1}{\sin \theta} \)
• Secant (\( \sec \theta \)): \( \sec \theta = \frac{1}{\cos \theta} \)
• Cotangent (\( \cot \theta \)): \( \cot \theta = \frac{1}{\tan \theta} \)

These functions are helpful in solving equations where the main trigonometric functions are inconvenient to use themselves. Knowing these relationships enables a smoother transition between using trigonometric functions and their reciprocals.

Sine Function

The sine function is a cornerstone of trigonometry. It is defined as the ratio of the length of the side opposite the angle \( \theta \) to the length of the hypotenuse in a right-angle triangle. Mathematically, it is written as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

• The sine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \, \text{radians} \).
• Its range is between -1 and 1, inclusive.
• The function is positive in the first and second quadrants.

Understanding sine is critical because many trigonometric properties and identities revolve around it. In context, \( \csc 60^{\circ} \) being the reciprocal of \( \sin 60^{\circ} \) demonstrates the close interdependence of these functions.

Angle Conversion

Angle conversion is an essential skill when working with trigonometric functions, as calculations can be made in degrees or radians. To convert degrees to radians, the simple formula is:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)

For example, if you need to convert \( 60^{\circ} \) to radians, it would be \( 60 \times \frac{\pi}{180} = \frac{\pi}{3} \).
Conversely, to convert radians to degrees, use:

• \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

Converting angles appropriately is crucial in trigonometry problems, especially when the problem presents angles in one unit but expects answers in another.