Problem 34 - Algebra and Trigonometry

Question

Find a cofunction with the same value as the given expression. $$ \csc 35^{\circ} $$

Step-by-Step Solution

Answer

The cofunction with the same value as the given expression \(\csc 35^{\circ}\) is \(\sin 55^{\circ}\).

Step 1: Identifying the Cofunction and Complementary Angle

First, identify the cofunction of cosecant, which is sine. Then, find the complementary angle to \(35^{\circ}\), which is \(90^{\circ} - 35^{\circ} = 55^{\circ}\). Therefore the solution will be of the form \(\sin(55^{\circ})\).

Step 2: Verifying the Value

Verify that the sine of the complementary angle has the same value as the original cosecant function. Since sine and cosecant are cofunctions, they have the same value when their angles are complementary. So \(\sin(55^{\circ})\) will indeed have the same value as \(\csc(35^{\circ})\).

Key Concepts

Cosecant Function

Understanding the cosecant function is essential for anyone studying trigonometry. It's one of the six fundamental trigonometric functions, often overshadowed by its more commonly used sibling, the sine function. The cosecant function, denoted as \textbf{csc}, is defined as the reciprocal of sine. Thus, if you know the value of the sine function, finding the cosecant value is straight forward: \

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

Given an angle \( \theta \) in a right triangle, where \( \theta \) is not the right angle, the cosecant of \( \theta \) is the length of the hypotenuse divided by the length of the opposite side.

In the context of the exercise, we're looking at \( \text{csc} (35^{\text{o}}) \) which means we need to take the reciprocal of the sine of \( 35^{\text{o}} \). A deep understanding of this function allows students to maneuver effectively within trigonometric identities and effectively understand their graph representations as well.

Complementary Angles

Angles are complementary if their sum equals \( 90^{\text{o}} \), which is the angle of a perfect square corner. Whenever you see two angles described as complementary, imagine them fitting together to form a right angle. This characteristic is pivotal when dealing with cofunction identities in trigonometry.

Why are complementary angles important in trigonometry? They form the basis for understanding that some trigonometric functions of an angle have the same value as other trigonometric functions of its complementary angle — this is where the concept of cofunction comes into play. For example, as shown in the original exercise \( \text{sin}(55^{\text{o}}) = \text{csc}(35^{\text{o}}) \) since \( 55^{\text{o}} + 35^{\text{o}} = 90^{\text{o}} \). This understanding will be indispensable when solving various trigonometric problems and proofs.

Sine Function

The sine function is perhaps one of the most fundamental components of trigonometry, representing the ratio of the length of the side opposite to an angle to the length of the hypotenuse in a right-angled triangle. The sine of an angle \( \theta \) is commonly written as \( \text{sin}(\theta) \).

How is it calculated?

Simply by using the formula \( \text{sin}(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).

The sine function is particularly important because it describes a wave-like pattern, which can be visualized through the sine wave—a concept used widely in physics and engineering. Most importantly for our discussion, when paired with the concept of complementary angles, it's essential in understanding the cofunction identities within trigonometry. Which is exactly what is employed in the solution for the exercise question: by knowing that \( \text{sin}(35^{\text{o}}) \) is the cofunction of \( \text{csc}(35^{\text{o}}) \) with \( \text{sin}(55^{\text{o}}) \) complementing it.