Problem 24 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \sec (-\pi) $$

Step-by-Step Solution

Answer

The secant of \(-\pi\) radians is \(-1\).

Step 1: Understanding the secant function

The secant (sec) is a trigonometric function. It is defined as the reciprocal of another trigonometric function, the cosine (cos). This means that the sec of an angle is 1 divided by the cosine of that angle. In math terms, this would be \(sec(x) = \frac{1}{cos(x)}\).

Step 2: Understanding \(-\pi\) radians

Radians are another way to measure angles, just like degrees. The complete revolution around a circle is \(2\pi\) radians, which means a half revolution (or a straight line) is \(\pi\) radians. If the angle is negative like \(-\pi\), it means we are measuring the angle in the opposite direction.

Step 3: Calculating the secant of \(-\pi\) radians

Now we can substitute \(-\pi\) into the equation for sec. We know that \(cos(-\pi) = -1\) and the secant is the reciprocal of the cosine, therefore \(\sec(-\pi) = \frac{1}{cos(-\pi)} = -1\).

Key Concepts

Secant Function

The secant function is one of the lesser-known trigonometric functions, but it plays an important role in mathematics. When you hear 'secant', think of it as being the "opposite cousin" to the cosine function. Essentially, the secant of an angle is the reciprocal of the cosine of that angle. This means it flips the cosine value upside down. Here's the formula:

• For any angle \(x\), the secant is given by \(\sec(x) = \frac{1}{\cos(x)}\).

When calculating secant, remember you are looking for what number you must multiply by the cosine to get 1. It's like solving the equation \(\cos(x) \times a = 1\). If you find the cosine to be zero, the secant becomes undefined, because you can't divide by zero.
Unlike cosine which stays between -1 and 1, secant can go beyond those limits depending on the angle. That’s what makes it an interesting function. Keep this in mind when handling trigonometric functions!

Radian Measure

Radians are a way to measure angles without relying on arbitrary degree numbers. It’s all about the circle! A full circle measures \(2\pi\) radians, or 360 degrees. Therefore, half a circle, or a straight line, measures \(\pi\) radians, equating to 180 degrees.

• Mathematicians and scientists prefer using radians because they often make calculations simpler.

When an angle is negative, like \(-\pi\), it simply indicates that you are measuring the angle in the opposite direction. Think of it as moving clockwise around the circle instead of the usual counterclockwise direction.
Beyond calculations, understanding radians improves your grasp on circular and rotational dynamics across varying fields, from physics to even computer graphics. Don't let the symbol \(\pi\) intimidate you; it’s just another helpful tool like degrees, made for precision.

Reciprocal Function

A reciprocal function is not limited to trigonometry—it exists in basic algebra too. It's a mathematical operation that multiplies a number by its flipped version to get 1. The core idea is to find what number, when multiplied by the given number, equals 1.

• For a number \(a\), its reciprocal is \(\frac{1}{a}\).

When applied in trigonometry, you take a trig function like cosine, and the reciprocal becomes secant:

• Cosine \(\rightarrow\) Secant \(\sec(x) = \frac{1}{\cos(x)}\)

Remember, division by zero is undefined, so you cannot find the secant of an angle if its cosine is zero—like at 90 degrees or \(\frac{\pi}{2}\). The reciprocal concept is helpful in simplifying expressions and solving equations by transforming multiplicative problems into simpler additive ones.