Problem 57 - Algebra 2

Question

Writing Explain why each expression is undefined. $$ \sec 90^{\circ} $$

Step-by-Step Solution

Answer

Sec(90 degrees) is undefined because it equals 1 divided by cos(90 degrees), which equals 1 divided by 0, and division by zero in mathematics is undefined.

Step 1: Understanding secant

The secant of an angle is the reciprocal of the cosine of that angle. Therefore, when we see sec(90 degrees), it's basically 1 / cos(90 degrees).

Step 2: Calculating the cosine of 90 degrees

Next, we calculate the cosine of 90 degrees. From trigonometric ratios, we know that cos(90 degrees) = 0.

Step 3: Explaining why sec(90 degrees) is undefined

Now we apply the value of cos(90 degrees) back into the original equation sec(90 degrees) = 1 / cos(90 degrees). This gives us 1 / 0. In mathematics, division by zero is undefined hence making sec(90 degrees) undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, helping us describe relationships in right-angled triangles and periodic patterns in circles. Common trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. These functions are key in connecting angles to the ratios of the sides in a triangle, and they also extend to describe wave behaviors in the context of circles.

• Sine \(\sin\) represents the ratio of the opposite side to the hypotenuse.
• Cosine \(\cos\) represents the adjacent side to the hypotenuse ratio.
• Tangent \(\tan\) is the ratio of the opposite side to the adjacent side.

Each function has unique properties and is related through specific identities, as we'll explore in this article.

Reciprocal Identities

Reciprocal identities are handy relationships between trigonometric functions that express one function in terms of another. These identities help simplify complex calculations by allowing for alternate expressions.
The reciprocal identities are:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cot \theta = \frac{1}{\tan \theta}\)

For secant, this means that secant of an angle is just one over the cosine of that angle. Using these relationships can easily transform the way we solve equations and provide insight into undefined expressions.

Division by Zero

Division by zero is a mathematical expression that essentially breaks the rules. When we try to divide a number by zero, the operation doesn't yield a valid result, because there's no number that you can multiply by zero to get something other than zero.
Thus, any time you find yourself with a calculation like \(\frac{1}{0}\), it becomes undefined. Undefined expressions don't have a meaningful interpretation in mathematics.
This concept crucially informs why sec(90°) is undefined since it leads to division by zero when evaluated using reciprocal identities.

Secant Function

The secant function, denoted as \(\sec\), is one of the trigonometric functions and is specifically the reciprocal of cosine. It holds unique characteristics and undefined values at certain points due to its dependence on cosine.
For instance, \(\sec 90^\circ = \frac{1}{\cos 90^\circ}\). Since \(\cos 90^\circ = 0\), this results in division by zero. Thus, secant is undefined at 90° and similar angles where cosine equals zero.
Understanding these specifics helps avoid errors in calculations and gives better insights into the behaviors and limitations of the secant function.