Problem 25 - 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. $$t=0$$ (TABLE CAN'T COPY).

Step-by-Step Solution

Answer

The trigonometric functions at t=0 are: sin=0, cos=1, tan=0, csc=undefined, sec=1, cot=undefined.

Step 1: Define the Angle

Since the given value is \( t = 0 \), we are considering an angle of \(0\) radians. This corresponds to a position on the unit circle.

Step 2: Determine the Point on the Unit Circle

At \( t = 0 \) radians, the point on the unit circle is \((1, 0)\). This point will help us find the value of trigonometric functions at \( t = 0 \).

Step 3: Calculate Sine Function

The sine function is defined as the y-coordinate on the unit circle. For \( t = 0 \), \( \sin(0) = 0 \).

Step 4: Calculate Cosine Function

The cosine function is defined as the x-coordinate on the unit circle. For \( t = 0 \), \( \cos(0) = 1 \).

Step 5: Calculate Tangent Function

The tangent function is calculated as the ratio of the sine to the cosine. For \( t = 0 \), \( \tan(0) = \frac{0}{1} = 0 \).

Step 6: Calculate Cosecant Function

The cosecant function is the reciprocal of the sine function. Since \( \sin(0) = 0 \), \( \csc(0) \) is undefined.

Step 7: Calculate Secant Function

The secant function is the reciprocal of the cosine function. For \( t = 0 \), \( \sec(0) = \frac{1}{1} = 1 \).

Step 8: Calculate Cotangent Function

The cotangent function is the reciprocal of the tangent function. Since \( \tan(0) = 0 \), \( \cot(0) \) is undefined.

Key Concepts

Unit Circle

The unit circle is a fundamental tool used in trigonometry to understand the relationships between angles and trigonometric functions. As its name suggests, the unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle allows us to visualize trigonometric functions on the x-y plane. Every point on the unit circle corresponds to an angle measured in radians, as well as specific coordinate values \((x, y)\). These coordinates are crucial for calculating sine and cosine values:

• The x-coordinate of a point gives the cosine of the angle.
• The y-coordinate gives the sine of the angle.

For instance, at 0 radians, the point on the unit circle is (1, 0), which tells us that \(\cos(0) = 1\) and \(\sin(0) = 0\). The unit circle helps students to understand trigonometric functions as it translates abstract concepts into visual relationships. Furthermore, the unit circle facilitates the calculations of reciprocal functions, which we'll explore in more detail.

Radians

Radians offer an alternative to degrees for measuring angles, and they are often used in trigonometry due to their direct relationship with the unit circle. A full circle encompasses an angle of \(2\pi\) radians. Unlike degrees, where a circle is divided into 360 parts, radians are more straightforward and tie directly into the circle's geometry.The conversions are simple:

• \( \pi \) radians is equal to 180 degrees.
• Therefore, 1 radian is approximately 57.2958 degrees.
• An angle of 0 radians corresponds to 0 degrees, simply the starting point on the circle.

Using radians, we often deal with simple multiples of \( \pi \), like \( \pi/2 \) or \( \pi/4 \), which translate into recognizable angles such as 90 degrees and 45 degrees respectively. Radians make calculations in trigonometry simpler, especially when using the unit circle, since many calculations and formulas in calculus are much cleaner and intuitive with radian measure rather than degrees.

Reciprocal Functions

Reciprocal functions derive from the basic sine, cosine, and tangent functions. These functions flip or "reciprocate" the values of their respective primary functions, which can help explore additional angle properties and restrictions. When dealing with these functions, remember:

• Cosecant \( (\csc \theta) \): Reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \). At \( t = 0 \), \( \csc(0) \) is undefined because you cannot divide by zero (since \( \sin(0) = 0 \)).
• Secant \( (\sec \theta) \): Reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \). For \( t = 0 \), \( \sec(0) = \frac{1}{1} = 1 \).
• Cotangent \( (\cot \theta) \): Reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \). At \( t = 0 \), \( \cot(0) \) is undefined for the same reason as \( \csc \), \( \tan(0) = 0 \).

Understanding the concept of reciprocal functions helps identify where functions may become undefined and provides a deeper insight into the function behaviors at specific points and across different domains.