Problem 24 - Algebra and Trigonometry

Question

Evaluate the expression without using a calculator. $$\sin 30^{\circ} \csc 30^{\circ}$$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Understand the Definitions

The expression to evaluate is \( \sin 30^{\circ} \csc 30^{\circ} \). Here, \( \sin \) refers to the sine function, and \( \csc \) is the reciprocal function, cosecant, which is defined as \( \csc \theta = \frac{1}{\sin \theta} \).

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

From trigonometric knowledge, \( \sin 30^{\circ} = \frac{1}{2} \).

Step 3: Calculate \( \csc 30^{\circ} \)

Using the definition \( \csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} \), substitute the value of \( \sin 30^{\circ} \): \( \csc 30^{\circ} = \frac{1}{\frac{1}{2}} = 2 \).

Step 4: Multiply the Values

Now multiply \( \sin 30^{\circ} \) and \( \csc 30^{\circ} \): \( \sin 30^{\circ} \csc 30^{\circ} = \frac{1}{2} \times 2 = 1 \).

Step 5: Confirm the Result

Review the calculations to ensure accuracy. Since sine and cosecant are reciprocal functions, their product should yield 1, which confirms our answer is correct.

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \theta \), is a fundamental concept in trigonometry. It relates the angle \( \theta \) of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. For instance, the sine of 30 degrees is a commonly known value: \( \sin 30^{\circ} = \frac{1}{2} \). This shows that in a right triangle with a 30-degree angle, the side opposite this angle is half the length of the hypotenuse.

To understand the sine function better, remember:

• The sine values range between -1 and 1 for all angles.
• It is periodic with a period of 360 degrees or \( 2\pi \) radians.
• It is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).

Knowing these basic properties will help you solve many trigonometric problems efficiently.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six trigonometric functions. It is not as commonly used as sine or cosine, but it plays a crucial role in many trigonometric identities and equations. The cosecant is essentially the reciprocal of the sine function, defined as \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, it is crucial to note that whenever the sine of an angle is zero, the cosecant is undefined, because you cannot divide by zero.

Let's take 30 degrees as an example again. From the definition, you know: \( \csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{\frac{1}{2}} = 2 \). This calculation shows how the cosecant function helps in transforming a basic trigonometric function into its reciprocal form.

• The range of cosecant values does not include between -1 and 1.
• It is periodic, repeating every 360 degrees or \( 2\pi \) radians.
• It coincides with the existence of sine values, excluding zero.

Understanding cosecant expands your ability to solve and simplify trigonometric expressions effectively.

Reciprocal Identities

Reciprocal identities are fundamental in trigonometry. They connect trigonometric functions with each other. These identities state that each primary trigonometric function has a corresponding reciprocal function. For sine, the reciprocal is cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Similarly, other reciprocal pairs include cosine with secant, and tangent with cotangent.

In the given problem \( \sin 30^{\circ} \csc 30^{\circ} \), applying the reciprocal identity simplifies the expression quickly. By definition, since \( \csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} \), multiplying sin and cosecant of the same angle yields 1:
\[ \sin 30^{\circ} \times \csc 30^{\circ} = \sin 30^{\circ} \times \frac{1}{\sin 30^{\circ}} = 1 \]

Understanding these identities allows you to manipulate and simplify trigonometric expressions seamlessly. Such proficiency is invaluable for solving more complex trigonometric equations and understanding the relationships between different angles and their functions.