Problem 24 - Algebra and Trigonometry

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

\(\sin\left(\frac{\pi}{2}\right) = 1\), \(\cos\left(\frac{\pi}{2}\right) = 0\), \(\tan\left(\frac{\pi}{2}\right) = \text{undefined}\), \(\csc\left(\frac{\pi}{2}\right) = 1\), \(\sec\left(\frac{\pi}{2}\right) = \text{undefined}\), \(\cot\left(\frac{\pi}{2}\right) = 0\).

Step 1: Understand the Problem

We need to find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) at the angle \( t = \frac{\pi}{2} \) radians.

Step 2: Evaluate Sine Function

The sine function at \( t = \frac{\pi}{2} \) is \( \sin\left(\frac{\pi}{2}\right) = 1 \).

Step 3: Evaluate Cosine Function

The cosine function at \( t = \frac{\pi}{2} \) is \( \cos\left(\frac{\pi}{2}\right) = 0 \).

Step 4: Evaluate Tangent Function

The tangent function is the ratio of sine over cosine. At \( t = \frac{\pi}{2} \), \( \tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} \), which is undefined.

Step 5: Evaluate Cosecant Function

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

Step 6: Evaluate Secant Function

The secant function is the reciprocal of cosine. At \( t = \frac{\pi}{2} \), \( \sec\left(\frac{\pi}{2}\right) = \frac{1}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} \), which is undefined.

Step 7: Evaluate Cotangent Function

The cotangent function is the reciprocal of tangent. At \( t = \frac{\pi}{2} \), \( \cot\left(\frac{\pi}{2}\right) = \frac{1}{\tan\left(\frac{\pi}{2}\right)} = \frac{0}{1} = 0 \).

Key Concepts

Sine Function

The sine function, denoted as \( \sin \), is a fundamental trigonometric function that relates the angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. In the unit circle, \( \sin \) is the y-coordinate of the corresponding point on the circle.
The sine function is periodic with a period of \( 2\pi \), meaning \( \sin(\theta + 2\pi) = \sin(\theta) \) for any angle \( \theta \). At \( t = \frac{\pi}{2} \), the value of \( \sin \left( \frac{\pi}{2} \right) = 1 \). This reflects the topmost point on the unit circle where the y-coordinate is exactly 1.
Understanding the sine function's values at key angles like \( \frac{\pi}{2} \) can help solve various trigonometric equations and is crucial for studying wave functions and oscillations.

Cosine Function

The cosine function, represented by \( \cos \), helps find the ratio of the adjacent side over the hypotenuse in a right triangle. On the unit circle, \( \cos \) is the x-coordinate of the point associated with a particular angle.
Cosine, just like sine, is periodic but behaves differently in terms of symmetry. It is an even function, thus \( \cos(-\theta) = \cos(\theta) \). Its value cycles with a period of \( 2\pi \), making \( \cos(\theta + 2\pi) = \cos(\theta) \).
For \( t = \frac{\pi}{2} \), the cosine function calculates to \( \cos \left( \frac{\pi}{2} \right) = 0 \). This indicates that the x-coordinate on the unit circle at this angle is zero, placing the point directly above the origin.

Tangent Function

Tangent, or \( \tan \), is found by dividing the sine function by the cosine function: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function gives the ratio of the opposite side to the adjacent side in a right triangle.
Unlike sine and cosine, tangent is periodic with a period of \( \pi \), aligning with its properties like \( \tan(\theta + \pi) = \tan(\theta) \). It is undefined when \( \cos(\theta) = 0 \) due to division by zero.
At \( t = \frac{\pi}{2} \), \( \tan \left( \frac{\pi}{2} \right) \) is undefined because the cosine at this point is 0. This means you can't divide by zero, hence the tangent function doesn't exist for this angle.

Radian Measurement

Radian measurement is a way to express angles based on the radius of a circle. One radian is the angle created when the length of the arc is equal to the radius of the circle.
This measurement is closely related to many trigonometric functions and is more natural in advanced mathematics due to its direct relationship with the geometry of a circle. Unlike degrees, where a full circle is 360 degrees, in radians, a full circle is \( 2\pi \) radians.
For example, \( \frac{\pi}{2} \) radians corresponds to a 90-degree angle, marking the point where the angle sweeps out a quarter of the circle's circumference.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary trigonometric functions by reversing their ratios.
Here are the reciprocal functions:

• **Cosecant (\( \csc \))**: The reciprocal of sine, \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• **Secant (\( \sec \))**: The reciprocal of cosine, \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• **Cotangent (\( \cot \))**: The reciprocal of tangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

These functions become undefined when their base trigonometric function results in zero, as dividing by zero is impossible.
For \( t = \frac{\pi}{2} \):
The cosecant value at \( \frac{\pi}{2} \) is 1, as \( \sin \left( \frac{\pi}{2} \right) \) is 1.
The secant and cotangent are undefined due to division by zero because \( \cos \left( \frac{\pi}{2} \right) = 0 \), and \( \tan \left( \frac{\pi}{2} \right) \) is already undefined.