Problem 50 - Precalculus : Building Concepts and Connections

Question

Find a cofunction that has the same value as the given quantity. $$\csc 31^{\circ}$$

Step-by-Step Solution

Answer

The cofunction with the same value as \( \csc 31^{\circ} \) is \( \sec 59^{\circ} \).

Step 1: Identify the Angel 31 Degrees

Identify the given angle which is 31 degrees.

Step 2: Calculate Complementary Angle

Compute the complementary angle by subtracting the given angle from 90 because in a right angle triangle, the sum of two non-right angles is 90 degrees. Doing so, you get \(90 - 31 = 59\).

Step 3: Apply Cofunction Identity

Finally apply the cofunction identity. The cofunction identity for cosecant is secant of the complementary angle. Thus, \( \csc 31^{\circ} = \sec 59^{\circ} \).

Key Concepts

Cosecant

Cosecant, denoted as \( \text{csc} \), is one of the six primary trigonometric functions. It is the reciprocal of the sine function, which means \( \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \). Understanding cosecant is essential for grasping various properties of right triangles and trigonometric identities. In any right angle triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Consequently, cosecant provides the ratio of the length of the hypotenuse to the length of the opposite side.

When working with cosecant, it’s important to remember that it will be undefined whenever its corresponding sine value is zero, as division by zero is undefined in mathematics. The function reaches its maximums and minimums where sine reaches its minimums and maximums. Understanding this reciprocal relationship enhances the comprehension of trigonometric functions and their applications in various disciplines, such as physics, engineering, and architecture.

Complementary Angles

Complementary angles are a fundamental concept in geometry. Two angles are said to be complementary if their sum equals 90 degrees, forming a right angle when combined. In the context of trigonometry, the notion of complementary angles is crucial when dealing with certain trigonometric identities known as cofunction identities.

These identities exploit the relationship between the trigonometric functions of complementary angles. For example, as seen in the exercise, if we have an angle \( \theta \), its complement is \( 90^{\text{\circ}} - \theta \). Remember that in any right angle triangle, the sum of the angles must add up to 180 degrees. Therefore, once one of the angles is the right angle (90 degrees), the other two must be complementary.

Trigonometric Identities

Trigonometric identities are equalities that involve trigonometric functions and are true for all values of the occurring variables where both sides of the equality are defined. These identities are useful not only in geometry and algebra but also in calculus and beyond. Cofunction identities are a special type of trigonometric identities that express the relationship between the trigonometric functions of complementary angles.

One of the key cofunction identities states that \( \text{sin}(\theta) = \text{cos}(90^{\text{\circ}} - \theta) \), and the corresponding identity for cosecant, which our exercise involves, is \( \text{csc}(\theta) = \text{sec}(90^{\text{\circ}} - \theta) \). These identities are particularly helpful when finding the value of a trigonometric function at an unfamiliar angle by relating it to a function of a more familiar angle.

Right Angle Triangle

A right angle triangle is a triangle in which one of the angles is exactly 90 degrees. The two sides forming this angle are called the legs, and the side opposite the right angle is known as the hypotenuse. It is the longest side of the triangle. Right angle triangles are central to trigonometry.

In such a triangle, trigonometric functions can be defined as ratios of the triangle's sides. For example, the sine of an angle is the ratio between the length of the opposite side and the hypotenuse, and the cosine is the ratio between the adjacent side and the hypotenuse. Due to the nature of right triangles, the Pythagorean theorem is applicable, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides \( (a^2 + b^2 = c^2) \). This provides a valuable tool for solving many practical problems that involve right angle triangles.