Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \csc 48^{\circ} $$

Step-by-Step Solution

Answer

The function \( \csc(48^{\circ}) \) equals \( 1.3450 \) using its cofunction.

Step 1: Understanding Cofunctions

Cofunction identities relate trigonometric functions of complementary angles. The cofunction identity for cosecant is \(\csc(\theta) = \sec(90^{\circ} - \theta)\). This means the cosecant of an angle is equal to the secant of its complementary angle.

Step 2: Applying the Cofunction Identity

Using the identity \(\csc(\theta) = \sec(90^{\circ} - \theta)\), substitute \(\theta = 48^{\circ}\) to find: \(\csc(48^{\circ}) = \sec(90^{\circ} - 48^{\circ})\). Simplifying inside the parentheses, we have \(\sec(42^{\circ})\).

Step 3: Calculating the Function Value

Now calculate \(\sec(42^{\circ})\). Recall that \(\sec(x) = \frac{1}{\cos(x)}\), so we need \(\cos(42^{\circ})\). Using a calculator, find \(\cos(42^{\circ}) \approx 0.7431\). Thus, \(\sec(42^{\circ}) = \frac{1}{0.7431} \approx 1.3450\).

Step 4: Finalizing the Value

The final value for \(\csc(48^{\circ})\) via its cofunction is \(\sec(42^{\circ})\) which equals approximately \(1.3450\) rounded to four decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of triangles and have applications in various fields such as physics, engineering, and computer science. The primary trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function provides a different ratio of sides in a right-angled triangle.

The reciprocal functions include cosecant (\( \csc \theta \)), secant (\( \sec \theta \)), and cotangent (\( \cot \theta \)). These functions are crucial when working with equations and identities. Understanding them helps to gain deeper insights into the properties of angles and can solve complex geometrical problems. Here are the relationships:

• Cosecant is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \)
• Secant is the reciprocal of cosine:\( \sec \theta = \frac{1}{\cos \theta} \)
• Cotangent is the reciprocal of tangent:\( \cot \theta = \frac{1}{\tan \theta} \)

With these functions, one can establish relations known as cofunction identities, which connect trigonometrically complementary angles.

Complementary Angles

Complementary angles, by definition, are two angles that add up to \( 90^{\circ} \). These angles are a cornerstone in understanding trigonometric identities and cofunction identities in particular. In trigonometry, when two angles are complementary, the trigonometric function of one angle is the cofunction of the other.

This concept is easy to remember: if you know the sine of one angle, then the cosine of its complement will equal that sine, because in any right triangle, the two non-right angles must be complementary.

• \( \sin(\theta) = \cos(90^{\circ} - \theta) \)
• \( \cos(\theta) = \sin(90^{\circ} - \theta) \)
• \( \tan(\theta) = \cot(90^{\circ} - \theta) \)
• \( \csc(\theta) = \sec(90^{\circ} - \theta) \)
• \( \sec(\theta) = \csc(90^{\circ} - \theta) \)
• \( \cot(\theta) = \tan(90^{\circ} - \theta) \)

The importance of complementary angles lies not only in solving trigonometric equations but also in simplifying expressions to make calculations easier.

Secant Function

The secant function (\( \sec \theta \)) is one of the six fundamental trigonometric functions. It's defined as the reciprocal of the cosine function. Understanding secant is crucial because it has unique properties and applications in solving problems with trigonometric identities and functions. When you express the secant of an angle, it connects directly to its cosine counterpart. Mathematically, it’s given by: \( \sec \theta = \frac{1}{\cos \theta} \).

The secant function helps find angles in various problems by connecting them through the cosine's value. If you consider angles within their respective quadrants, secant, like other trigonometric functions, has both positive and negative values. It is undefined whenever the cosine equals zero, as dividing by zero does not yield a real number. This will occur at angles where:

• \( \cos \theta = 0 \), usually multiples of \( 90^{\circ} \) or\( 270^{\circ} \)

In this exercise, the secant function plays a vital role in finding the cofunction of \( \csc(48^{\circ}) \), using the identity \( \sec(42^{\circ}) \), by understanding these relationships this problem is easily tackled.