Problem 12 - Precalculus

Question

Use the definitions (not a calculator) to evaluate the six trigonometric functions of each angle. If a value is undefined, state this. $$3 \pi / 2$$

Step-by-Step Solution

Answer

Sine: -1, Cosine: 0, Tangent: Undefined, Cosecant: -1, Secant: Undefined, Cotangent: 0.

Step 1: Understanding the Angle

The given angle is \( \frac{3\pi}{2} \) radians. This angle corresponds to \(270^{\circ}\) as a full circle is \(2\pi\) radians or \(360^{\circ}\). This places \( \frac{3\pi}{2} \) radians on the negative y-axis.

Step 2: Determine the Sine of the Angle

Sine of an angle is the y-coordinate of the point on the unit circle. For \( \frac{3\pi}{2} \), the point is (0, -1). Hence, \( \sin(\frac{3\pi}{2}) = -1 \).

Step 3: Determine the Cosine of the Angle

Cosine of an angle is the x-coordinate of the point on the unit circle. For \( \frac{3\pi}{2} \), the point is (0, -1). Hence, \( \cos(\frac{3\pi}{2}) = 0 \).

Step 4: Determine the Tangent of the Angle

Tangent of an angle is the ratio of the sine to the cosine. So, \( \tan(\frac{3\pi}{2}) = \frac{-1}{0} \), which is undefined because division by zero is not possible.

Step 5: Determine the Cosecant of the Angle

Cosecant is the reciprocal of sine. Since \( \sin(\frac{3\pi}{2}) = -1 \), then \( \csc(\frac{3\pi}{2}) = \frac{1}{-1} = -1 \).

Step 6: Determine the Secant of the Angle

Secant is the reciprocal of cosine. Cosine is 0 at \( \frac{3\pi}{2} \), thus \( \sec(\frac{3\pi}{2}) = \frac{1}{0} \), which is undefined.

Step 7: Determine the Cotangent of the Angle

Cotangent is the reciprocal of tangent. Since the tangent is undefined, the cotangent, \( \cot(\frac{3\pi}{2}) = \frac{0}{-1} = 0 \).

Key Concepts

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It's a circle centered at the origin (0,0) of the coordinate plane with a radius of 1. This simple setup makes it easy to find the sine and cosine of any angle.

On the unit circle, the position of any angle is represented by a point (x, y). Here:

• x is the cosine of the angle, and
• y is the sine of the angle.

For example, when an angle measures \( \frac{3\pi}{2} \text{ radians} \), which is equivalent to \( 270^{\circ}\), the corresponding point on the unit circle is (0, -1).

Understanding the unit circle helps you visually grasp the values of \( ext{sine} \) and \( ext{cosine} \) at different angles around the circle. It serves as a guide for evaluating trigonometric functions, noting where they are defined or undefined.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate angles to the coordinates on the unit circle.

Sine (\( ext{sin} \)) measures the vertical component ({y-coordinate}) of an angle's corresponding point on the unit circle.

• For \( \frac{3\pi}{2} \), the sine value is -1.

Cosine (\( ext{cos} \)) measures the horizontal component ({x-coordinate}) of the point.

• At \( \frac{3\pi}{2} \), the cosine value is 0.

These functions depend on the angle's position within the circle and describe how far it extends along the vertical and horizontal axes. Recognizing these relationships provides the foundation for other more complex trigonometric functions.

Angle Measurement

Angles can be measured in degrees and radians, which are both vital for trigonometry. A full circle measures \( 360^{\circ} \text{ or } 2\pi \text{ radians}.\)

When working with radians, we often use \( \pi \\), which roughly equals 3.14, to simplify expressions of angles. For instance, \( \frac{3\pi}{2} \) radians is a straightforward expression for \( 270^{\circ}\).

Radian measurement is particularly useful in calculus and other higher branches of mathematics, where the relationship between angles and arc lengths is more precisely expressed.

Reciprocal Functions

Reciprocal functions add another layer to understanding trigonometry. These functions are derived by taking the reciprocal of sine, cosine, and tangent.

Cosecant (\( ext{csc} \)): As the reciprocal of sine, \( ext{csc}(\theta) = rac{1}{ ext{sin}(\theta)}\). For \( \frac{3\pi}{2}\), \( ext{sin} \) is -1, so \( ext{csc} \left(\frac{3\pi}{2}\right) = -1 \).

• It becomes undefined wherever sine is zero.

Secant (\( ext{sec} \)): The reciprocal of cosine, \( ext{sec}(\theta) = rac{1}{ ext{cos}(\theta)}\). Cosine is zero at \( \frac{3\pi}{2} \), making \( ext{sec} \left(\frac{3\pi}{2}\right) \) undefined.

• Secant is undefined wherever cosine is zero.

Cotangent (\( ext{cot} \)): This is the reciprocal of tangent, \( ext{cot}(\theta) = rac{1}{ ext{tan}(\theta)}\).

• Because tangent is already undefined at \(\frac{3\pi}{2}\), \( ext{cot} \left(\frac{3\pi}{2}\right) = 0.\)

Reciprocal functions enhance our deep understanding of trigonometric behavior, especially at angles where standard functions are undefined.