Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \csc 30^{\circ} $$

Step-by-Step Solution

Answer

\( \csc 30^{\circ} = 2 \).

Step 1: Understand the Function

The function \( \csc \theta \) is the cosecant of angle \( \theta \). Cosecant is the reciprocal of the sine function. So, \( \csc \theta = \frac{1}{\sin \theta} \). Our task is to find \( \csc 30^{\circ} \).

Step 2: Find \( \sin 30^{\circ} \)

We know from trigonometric values that \( \sin 30^{\circ} = \frac{1}{2} \). This is a standard angle value and its sine is easy to remember.

Step 3: Compute \( \csc 30^{\circ} \) using \( \sin 30^{\circ} \)

Since \( \csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} \), substitute \( \sin 30^{\circ} = \frac{1}{2} \): \[ \csc 30^{\circ} = \frac{1}{\frac{1}{2}} = 2 \]

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six fundamental trigonometric functions. It is particularly important because it provides a relationship between the angles and sides of a right-angled triangle. The cosecant of an angle is defined as the reciprocal of the sine of that angle. This means that if you know the sine of an angle, you can easily find its cosecant by taking the inverse. The mathematical expression for this relationship is \( \csc \theta = \frac{1}{\sin \theta} \).

• Cosecant function is undefined at angles where the sine is zero, such as 0°, 180°, 360°, etc.
• It is useful in solving problems where the side opposite the angle is known.

By understanding the cosecant function's relationship with the sine function, it becomes much easier to handle trigonometric problems efficiently and accurately.

Sine Function

The sine function is one of the most well-known trigonometric functions, symbolized as \( \sin \theta \). It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle for a given angle \( \theta \). The sine function is also one of the periodic functions, meaning it repeats values at specific intervals. It has a range between -1 and 1.

• Common angles have known sine values, like \( \sin 30^{\circ} = \frac{1}{2} \).
• The sine function is crucial in determining wave patterns and oscillatory motions.

In the context of our problem, knowing \( \sin 30^{\circ} = \frac{1}{2} \) is essential for using reciprocal identities to find the cosecant.

Reciprocal Identities

Reciprocal identities are essential for interconverting between different trigonometric functions. These identities state that certain trigonometric functions are reciprocals of each other. For instance, the cosine is the reciprocal of the secant, and the tangent is the reciprocal of the cotangent. In our case, the sine and cosecant functions are reciprocal.

• The reciprocal identity for sine and cosecant is \( \csc \theta = \frac{1}{\sin \theta} \).
• Using reciprocal identities can simplify many trigonometry problems.

These identities are extremely useful in evaluating trigonometric expressions and solving equations. They offer a straightforward method to switch between functions to find unknown values, such as finding \( \csc 30^{\circ} \) when \( \sin 30^{\circ} \) is known.