Problem 120 - Precalculus

Question

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\sec 270^{\circ}$$

Step-by-Step Solution

Answer

\( \sec 270^{\circ} \) results in division by zero, which is undefined, so it's an error.

Step 1: Convert Secant to Cosine

The secant function can be rewritten in terms of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we need to evaluate \( \frac{1}{\cos 270^{\circ}} \).

Step 2: Calculate Cosine at 270 Degrees

Determine the value of \( \cos 270^{\circ} \). In the unit circle, \( \cos 270^{\circ} = 0 \) since at 270 degrees, the x-coordinate of the point is 0.

Step 3: Evaluate the Secant Expression

Now, substitute \( \cos 270^{\circ} = 0 \) into the expression. You get \( \sec 270^{\circ} = \frac{1}{0} \). This results in division by zero, which is undefined in mathematics.

Step 4: Explain Error

The calculation resulted in undefined behavior because division by zero does not have a finite value. Hence, the calculator will show an error, as it cannot represent an undefined quantity.

Key Concepts

Secant

The secant function is a trigonometric function related closely to the cosine function. Secant is represented as \( \sec \theta \) and is defined as the reciprocal of the cosine function: \[ \sec \theta = \frac{1}{\cos \theta} \]This means that for any angle \( \theta \), you can find the secant by taking the value of the cosine and finding its reciprocal.

• If \( \cos \theta = 0 \), secant becomes undefined, because division by zero is undefined.
• The secant function is defined for all angles except where the cosine equals zero.

When calculating secant, always be mindful of the angle's cosine value.

Cosine

The cosine function is one of the basic trigonometric functions. It is denoted by \( \cos \theta \). The cosine of an angle in a right triangle is the ratio of the adjacent side over the hypotenuse.

• In the unit circle, cosine reflects the x-coordinate of a point.
• Knowing the cosine allows you to find other functions like secant and sine.

For angle \( 270^{\circ} \), the unit circle tells us that \( \cos 270^{\circ} = 0 \). This is because the point at \( 270^{\circ} \) is located on the y-axis, where the x-coordinate, or cosine, is zero.

Unit Circle

The unit circle is an essential tool in trigonometry used to define trigonometric functions for all real numbers. It is a circle with a radius of 1 and is centered at the origin of a coordinate plane.

• The angle in the unit circle starts on the positive x-axis and rotates counter-clockwise.
• The coordinates of any point on the unit circle directly correspond to the values of the cosine (x-coordinate) and sine (y-coordinate) for that angle.

At \( 270^{\circ} \), the unit circle point is \((0, -1)\). This indicates that \( \cos 270^{\circ} = 0 \), leading to an undefined secant at this angle.

Undefined Behavior

In mathematics, undefined behavior refers to operations that do not result in a well-defined outcome. A prime example of this is division by zero.

• When a number is divided by zero, the result is undefined, as it does not produce a finite or meaningful number.
• In trigonometry, this often occurs with functions like secant, which involve division by the cosine function.

As seen in our example with \( \sec 270^{\circ} \), the calculation \( \sec 270^{\circ} = \frac{1}{\cos 270^{\circ}} = \frac{1}{0} \) results in undefined behavior, prompting an error on calculators.