Problem 27 - Precalculus Mathematics for Calculus

Question

Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t\). Use your answers to complete the table. $$t=\pi$$ (TABLE CAN'T COPY).

Step-by-Step Solution

Answer

At \( t = \pi \): \( \sin(\pi) = 0 \), \( \cos(\pi) = -1 \), \( \tan(\pi) = 0 \), \( \csc(\pi) \) undefined, \( \sec(\pi) = -1 \), \( \cot(\pi) \) undefined.

Step 1: Understanding the Question

We need to find the values of the six trigonometric functions at the given angle \( t = \pi \). These functions are sine, cosine, tangent, cosecant, secant, and cotangent.

Step 1: Sine Function \( \sin(\pi) \)

The sine of \( \pi \) is evaluated using the unit circle. For \( t = \pi \), the point is \((-1, 0)\). The sine value is the y-coordinate, hence \(\sin(\pi) = 0\).

Step 2: Cosine Function \( \cos(\pi) \)

The cosine is the x-coordinate of the point on the unit circle at \( t = \pi \). The point is \((-1, 0)\). Therefore, \(\cos(\pi) = -1\).

Step 3: Tangent Function \( \tan(\pi) \)

Tangent is the sine divided by the cosine. For \( \tan(\pi) = \frac{\sin(\pi)}{\cos(\pi)} = \frac{0}{-1} = 0\).

Step 4: Cosecant Function \( \csc(\pi) \)

Cosecant is the reciprocal of sine. Since \(\sin(\pi) = 0\), \(\csc(\pi)\) is undefined because division by zero is not possible.

Step 5: Secant Function \( \sec(\pi) \)

Secant is the reciprocal of cosine. \(\sec(\pi) = \frac{1}{\cos(\pi)} = \frac{1}{-1} = -1\).

Step 6: Cotangent Function \( \cot(\pi) \)

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

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry, used to define the sine, cosine, and tangent of an angle. Think of it as a circle with a radius of 1, centered at the origin (0,0) in the coordinate plane. Each point on the unit circle represents the coordinates (x, y) of the angle from the positive x-axis. This relationship is expressed as \( x = \cos(\theta) \) and \( y = \sin(\theta) \), where \( \theta \) is the angle in question.

In the case of \( t = \pi \):

• The point on the unit circle is (-1,0).
• The x-coordinate is -1, which gives us the cosine of \( \pi \).
• The y-coordinate is 0, which represents the sine of \( \pi \).

These coordinates help evaluate trigonometric functions for many angles. Remember, the unit circle helps visualize and find values for angles, making it easier to solve trigonometric problems.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basics: sine, cosine, and tangent. The three reciprocal functions are cosecant, secant, and cotangent, each the inverse or reciprocal of a primary function.

• Cosecant (\(\csc\)): The reciprocal of sine, defined as \(\csc(\theta) = \frac{1}{\sin(\theta)}\). For \( t = \pi \), since \( \sin(\pi) = 0 \), \( \csc(\pi) \) is undefined because dividing by zero is impossible.
• Secant (\(\sec\)): The reciprocal of cosine, expressed as \(\sec(\theta) = \frac{1}{\cos(\theta)}\). At \( t = \pi \), since \( \cos(\pi) = -1 \), \( \sec(\pi) = -1 \).
• Cotangent (\(\cot\)): The reciprocal of tangent, given by \(\cot(\theta) = \frac{1}{\tan(\theta)}\). The tangent of \( \pi \) is zero, so \( \cot(\pi) \) is undefined.

Understanding these reciprocals expands our trigonometry toolbox, even though care must be taken with undefined values.

Angle in Radians

Angles in radians provide a natural and convenient way of measuring angles based on the radius of a circle. A full circle is \(2\pi\) radians, which links a radian to the unit circle.

• Radian measure shows the angle corresponding to the arc's length equal to the radius of the circle.
• \( \pi \) radians, therefore, represent a straight angle or 180 degrees.

Using radians aligns with the circle's arc length, aiding in calculations involving periodic functions like sine and cosine. When \( t = \pi \), it places us at (-1, 0) on the unit circle, corresponding perfectly to this understanding as it encompasses half of the circle's circumference. This perspective makes radians integral in trigonometry, allowing a seamless switch between angle measures and practical applications on the unit circle.