Problem 62 - Contemporary Precalculus

Question

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\cos t}{\cos (t-\pi / 2)}=\cot t$$

Step-by-Step Solution

Answer

Question: Verify if the equation, $$\frac{\cos t}{\cos (t-\pi / 2)} = \cot t$$, could be an identity or not by graphing both sides of the equation. Answer: After evaluating both sides of the equation, we found that $$\frac{\cos t}{\sin t} = \frac{\cos t}{\sin t}$$. Given that the left-hand side and right-hand side of the equation are equal, the equation is an identity. Therefore, graphing both sides of the equation would result in the same graph, confirming that it is an identity.

Step 1: Evaluate Left Hand Side (LHS)

To evaluate the LHS, $$\frac{\cos t}{\cos (t-\pi / 2)},$$ we need to recall the trigonometric identity, $$\cos (t - \pi / 2) = \sin t$$. Therefore, the LHS becomes $$\frac{\cos t}{\sin t}$$.

Step 2: Evaluate Right Hand Side (RHS)

The RHS is already given in terms of a trigonometric function, $$\cot t$$. 2. Recall the definition of cotangent function

Step 3: Cotangent Function Definition

We need to recall the definition of the cotangent function, which is $$\cot t = \frac{\cos t}{\sin t}$$. 3. Compare LHS and RHS

Step 4: Compare LHS and RHS

We can now compare the LHS and RHS of the equation: $$ \frac{\cos t}{\cos (t-\pi / 2)} = \cot t \Rightarrow \frac{\cos t}{\sin t} = \frac{\cos t}{\sin t}. $$ 4. Conclusion

Step 5: Conclusion

As the LHS is equal to the RHS, the given equation, $$\frac{\cos t}{\cos (t-\pi / 2)}=\cot t$$, is an identity. Graphing both sides of the equation would result in the same graph, confirming that it is an identity.

Key Concepts

Cotangent Function

Understanding the cotangent function is fundamental when diving into trigonometric equations and identities. The cotangent is one of the six primary trigonometric functions and is the reciprocal of the tangent function. This implies that if you're aware of the tangent of an angle, you can easily find its cotangent by taking the reciprocal.

More formally, the cotangent of an angle, denoted as \( \cot \theta \), is the ratio of the adjacent side to the opposite side in a right-angled triangle. However, in terms of unit circle or function definitions, it is given by \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Graphically, the cotangent function has a period of \(\pi\), meaning it repeats its values every \(\pi\) radians. It also has vertical asymptotes where the sine function has zeros, because at those points, we are dividing by zero which is undefined, dramatically affecting the graph of the function. Recognizing this pattern is helpful when comparing trigonometric expressions or proving identities.

Graphical Analysis of Equations

Graphical analysis is a technique often used to visualize and solve equations, especially when dealing with trigonometric functions. It involves plotting the functions on a coordinate system and observing the intersections, behaviors, and properties of the graphs.

An identity in trigonometry is an equation that is true for all values of the variable, within the domain of the functions involved. To graphically determine if an equation is an identity, you plot both sides of the equation on the same axes.

When you have an identity, such as in the exercise where \( \frac{\cos t}{\cos (t-\pi / 2)} = \cot t \), if graphed correctly, you will observe that both sides of the equation would produce the exact same graph. This visual representation reinforces the analytical solution and provides a deeper comprehension of how the trigonometric functions are related.

Trigonometric Functions

Trigonometric functions form the cornerstone of trigonometry. They relate the angles of a triangle to the lengths of its sides and are extensively applied in science, engineering, and mathematics. The main trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Each function has unique properties and graphs that describe their behavior over an interval. For example, sine and cosine functions are periodic with a period of \(2\pi\), while tangent and cotangent functions have periods of \(\pi\). This periodicity means that they repeat their values over regular intervals, a crucial concept when solving equations or proving identities.

Studying the behavior of these functions, such as their symmetries, asymptotes, and intercepts, is essential for understanding more complex relations and equations in trigonometry. For students to deeply understand these concepts, visual aids like unit circles and function graphs can be invaluable, making it easier to internalize and apply the information.