Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the identity. $$\sec \left(90^{\circ}-\theta\right)=\square$$

Step-by-Step Solution

Answer

The identity is completed as: \(\sec \left(90^{\circ}-\theta\right)= \csc(\theta)\)

Step 1: Understanding the secant function

Recognize that the secant function is the reciprocal of the cosine function. That is, \(\sec(\theta) = \frac{1}{\cos(\theta)}\). So, \(\sec(90^{\circ} - \theta) = \frac{1}{\cos(90^{\circ} - \theta)}\)

Step 2: Using the co-function identities

Using the co-function identities, which state that the cosine of an angle is the sine of its complementary angle, the expression can be written as: \(\frac{1}{\sin(\theta)}\)

Step 3: Understanding the cosecant function

Recognize that \(\frac{1}{\sin(\theta)}\), is the cosecant function, denoted by \(\csc(\theta)\). This is our final answer.

Key Concepts

Secant Function

The secant function, denoted as \( \text{sec} \theta \), plays a critical role in trigonometry and often appears in various mathematical problems. It is closely related to the cosine function, one of the primary trigonometric functions. Consider the unit circle where the angle \( \theta \) corresponds to a point on the circle's circumference. The cosine of \( \theta \) describes the x-coordinate of this point, and the secant is simply the reciprocal of this value. In mathematical terms,
\[ \text{sec}(\theta) = \frac{1}{\text{cos}(\theta)} \]
This relationship allows us to transform between cosine and secant functions easily. Understanding this is essential because it serves as the foundation for more advanced trigonometric manipulations, including the solution to our original exercise.

Co-function Identities

Co-function identities are a group of trigonometric formulas that highlight an intriguing symmetry in right-angled triangles. In these identities, the trigonometric function of an angle is equal to the co-function of its complement. The complement of an angle \( \theta \) is \( 90^{\text{o}} - \theta \). One of the most used co-function identities is:
\[ \text{cos}(\theta) = \text{sin}(90^{\text{o}} - \theta) \]
These identities are particularly useful when dealing with complementary angles, as seen in the given exercise. By transforming the secant of an angle into the cosecant of its complement, we can solve problems that at first glance seem to involve unknown identities. Co-function identities also reveal the deep interconnectedness of trigonometric functions and are a testament to the elegance of mathematical symmetry.

Cosecant Function

The cosecant function, often symbolized as \( \text{csc}(\theta) \), is less commonly used than its counterparts like sine or cosine, but equally important. It is defined as the reciprocal of the sine function:
\[ \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \]
Imagine a right-angled triangle where \( \theta \) is one of the non-right angles. Sine represents the ratio of the length of the side opposite to \( \theta \) over the hypotenuse. Cosecant, on the other hand, gives us the ratio of the hypotenuse to the length of the opposite side. In the context of the exercise, understanding that cosecant is the 'flip' of the sine function reveals the solution to the identity problem when applying the co-function identities to the secant function.