Problem 15 - Precalculus Essentials

Question

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\cot \frac{\pi}{2}$$

Step-by-Step Solution

Answer

The cotangent function at \(\frac{\pi}{2}\) is 0.

Step 1: Definition of cotangent

First, recall the definition of the cotangent of angle \(x\), which is \(\cot(x)=\frac{1}{\tan(x)}\), or equivalently, \(\cot(x)=\frac{\cos(x)}{\sin(x)}\).

Step 2: Evaluate the denominator

Then, check the denominator of the function - since cotangent is the reciprocal of the tangent, the cotangent function will be undefined where the sine (in the denominator) equals zero. Check \(\sin(\frac{\pi}{2})\). We can remember that at \(\frac{\pi}{2}\) (or 90 degrees), \(\sin\) is at its maximum of 1.

Step 3: Evaluate the numerator

Next, check the numerator of the cotangent function. For \(\cos(\frac{\pi}{2})\), we remember that at \(\frac{\pi}{2}\) (or 90 degrees), \(\cos\) equals zero.

Step 4: Compute the cotangent

Now compute the cotangent function at \(\frac{\pi}{2}\) by substituting the values from step 2 and 3 into the formula: \(\cot(\frac{\pi}{2}) = \frac{0}{1}\).

Key Concepts

Understanding Cotangent

The cotangent, often abbreviated as "cot," is a key trigonometric function closely related to tangent. Defining it is simple:

• It is the reciprocal of the tangent. This means \[\cot(x) = \frac{1}{\tan(x)}\]
• It can also be expressed as the ratio of cosine to sine: \[\cot(x) = \frac{\cos(x)}{\sin(x)}\]

The cotangent function is used to relate the angles and sides in a right triangle. It becomes undefined when the sine of an angle is zero (because division by zero is not possible). Pay attention to special angles where this might occur to understand why the function might be undefined at certain points.

Exploring Quadrantal Angles

Quadrantal angles are special angles that lie on the x-axis or y-axis of the coordinate plane. These angles are important in trigonometry because the sine and cosine values for these angles are straightforward.

• These angles are typically 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\) radians (or 0°, 90°, 180°, and 270°).
• At quadrantal angles, one of the trigonometric functions will often be zero.

Understanding these angles helps in quickly assessing the behavior of trigonometric functions without extensive calculations. This is particularly useful in simplifying trigonometric expressions or solving equations.

Identifying Undefined Expressions

In the world of trigonometry, certain expressions can become undefined. This occurs when there is a division by zero in any of the trigonometric functions' expressions.
For cotangent, expressed as \[\frac{\cos(x)}{\sin(x)}\] the expression becomes undefined when \(\sin(x)\) is zero because division by zero is undefined in mathematics.

• Common points of undefined behavior for trigonometric functions include 0, \(\pi\), and \(2\pi\) radians, where each rotation yields a sine of zero.
• This is why it's essential to check the denominator before concluding a function value.

Knowing these points can save time and prevent errors when evaluating trigonometric functions.

Introducing Sine in Trigonometry

Sine is one of the foundational trigonometric functions. It helps measure the vertical component of an angle in a unit circle and is defined simply as:
The opposite side over the hypotenuse in a right triangle. Algebraically, it is expressed for angle \(x\) in radians as:\[\sin(x)\] Sine values range between -1 and 1. At specific angles such as \(\frac{\pi}{2}\), sine reaches its maximum absolute value. Understanding sine helps in forming sinusoidal functions, which model periodic phenomena in science and engineering.

The Role of Cosine

Cosine works in tandem with sine in the realm of trigonometric functions. It describes the horizontal component of an angle on the unit circle. Cosine for angle \(x\) is mathematically defined as:\[\cos(x)\] In a right triangle, it’s the ratio of the adjacent side to the hypotenuse. Cosine values, like sine, range from -1 to 1, but unlike sine, cosine starts at 1 for an angle of 0 and decreases to zero by \(\frac{\pi}{2}\).

• At \(\frac{\pi}{2}\), cosine is zero, making it crucial in determining where certain trigonometric functions are undefined.
• Understanding cosine helps illuminate wave patterns and signal behaviors in physics and engineering.

Mastering these concepts enables the resolution of complex trigonometric problems and equations.