Problem 26 - 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. $$t=\frac{\pi}{2}$$ (TABLE CAN'T COPY).

Step-by-Step Solution

Answer

Sine = 1, Cosine = 0, Tangent = undefined, Cosecant = 1, Secant = undefined, Cotangent = 0.

Step 1: Understanding the angle

We start by understanding that the angle given is in radians. The angle is \( t = \frac{\pi}{2} \), which is equivalent to 90 degrees in the unit circle.

Step 2: Sine function value

At \( t = \frac{\pi}{2} \), the sine function, \( \sin(t) \), represents the y-coordinate of the corresponding point on the unit circle. At \( 90^\circ \), this point is \((0, 1)\), so \( \sin\left(\frac{\pi}{2}\right) = 1 \).

Step 3: Cosine function value

For the cosine function, \( \cos(t) \), it represents the x-coordinate on the unit circle. At \( 90^\circ \), the point is \((0, 1)\), so \( \cos\left(\frac{\pi}{2}\right) = 0 \).

Step 4: Tangent function value

The tangent function is defined as \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). Substituting the sine and cosine values, we get \( \tan\left(\frac{\pi}{2}\right) = \frac{1}{0} \), which is undefined because division by zero is not possible.

Step 5: Cosecant function value

The cosecant function is the reciprocal of sine, \( \csc(t) = \frac{1}{\sin(t)} \). Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), \( \csc\left(\frac{\pi}{2}\right) = 1 \).

Step 6: Secant function value

The secant function is the reciprocal of cosine, \( \sec(t) = \frac{1}{\cos(t)} \). With \( \cos\left(\frac{\pi}{2}\right) = 0 \), \( \sec\left(\frac{\pi}{2}\right) \) is undefined because division by zero is not possible.

Step 7: Cotangent function value

The cotangent function is the reciprocal of tangent, \( \cot(t) = \frac{1}{\tan(t)} \). Since \( \tan\left(\frac{\pi}{2}\right) \) is undefined, \( \cot\left(\frac{\pi}{2}\right) = 0 \) based on the relation with sine and cosine.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, crucial for understanding angles and their corresponding points. Imagine a circle with a radius of one unit, centered at the origin of a coordinate plane. This circle allows us to easily visualize trigonometric functions. Each angle, measured in radians or degrees, corresponds to a specific point on this circle. For example:

• At 0 radians (or 0 degrees), we're at point (1, 0).
• At \(\frac{\pi}{2}\) radians (or 90 degrees), we find ourselves at (0, 1).
• At \(\pi\) radians (or 180 degrees), the point is (-1, 0).
• At \(\frac{3\pi}{2}\) radians (or 270 degrees), the point lies at (0, -1).
• And finally, at \(2\pi\) radians (or 360 degrees), we return to (1, 0).

These coordinates help us understand the values of sine and cosine, as they represent the y and x values on the circle respectively.

Radians

Radians offer a natural way to measure angles in mathematics and are defined by the arc length of a circle. One radian is the angle created when the radius is wrapped along the arc length of the circle. In a full circle, there are \(2\pi\) radians. Thus, this measure is rooted in the relationship between a circle's circumference and its radius. To convert between degrees and radians, use the conversion factor \(\pi\) radians equals 180 degrees. For example:

• 90 degrees converts to \(\frac{\pi}{2}\) radians.
• 180 degrees equals \(\pi\) radians.
• 360 degrees becomes \(2\pi\) radians.

Using radians simplifies many mathematical equations and analyses, particularly in calculus and physics. They enable easy calculation of arc lengths and area, making them indispensable in advanced mathematics.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are based on the basic sine, cosine, and tangent functions, extending their utility by considering their inverses. These include:

• Cosecant (\(\csc(t)\)), which is the inverse of sine: \(\csc(t) = \frac{1}{\sin(t)}\).
• Secant (\(\sec(t)\)), the inverse of cosine: \(\sec(t) = \frac{1}{\cos(t)}\).
• Cotangent (\(\cot(t)\)), which inversely relates to tangent: \(\cot(t) = \frac{1}{\tan(t)}\).

These functions allow for the reversal of relationships and offer flexibility in solving various trigonometric identities and equations. They can also help us determine angle sizes and side lengths in triangles when direct trigonometric ratios are not convenient.

Undefined Values in Trigonometry

In trigonometry, not all functions yield defined values for every angle. A function can become undefined primarily due to division by zero. When this occurs, it means the function lacks a finite value at that specific point.

• For example, tangent \(\tan(t) = \frac{\sin(t)}{\cos(t)}\) becomes undefined when \(\cos(t) = 0\), such as at \(\frac{\pi}{2}\) radians.
• The secant function (inverse cosine) \(\sec(t) = \frac{1}{\cos(t)}\) becomes undefined for similar reasons, like at \(\frac{\pi}{2}\) radians where cosine equals zero.
• Conversely, the cosecant \(\csc(t)\) and cotangent \(\cot(t)\) functions are undefined when \(\sin(t) = 0\), such as at 0 or \(\pi\) radians.

Recognizing where functions are undefined is crucial for correctly solving trigonometric equations and avoiding errors in calculations. Understanding these points ensures accurate interpretations in modeling and analyzing periodic phenomena.