Problem 31 - Algebra 2 and Trigonometry

Question

It is important to understand the underlying mathematics before using the calculator to solve trigonometric equations. For example, Adrian tried to use the intersect feature of his graphing calculator to find the solutions of the equation cot \(\theta=\sin \left(\theta-\frac{\pi}{2}\right)\) in the interval \(0 \leq \theta \leq \pi\) but got an error message. Follow the steps that Adrian used to solve the equation: (1) Enter \(Y_{1}=\frac{1}{\tan X}\) and \(Y_{2}=\sin \left(X-\frac{\pi}{2}\right)\) into the \(Y=\) menu. (2) Use the following viewing window to graph the equations: $$ X \min =0, \operatorname{Xmax}=\pi, X s c l=\frac{\pi}{6}, Y \min =-5, Y \max =5 $$ (3) The curves seem to intersect at \(\left(\frac{\pi}{2}, 0\right) .\) Press 2nd CALC 5 ENTER ENTER to select both curves. When the calculator asks for a guess, move the cursor near the intersection point using the arrow keys and then press ENTER a. Why does the calculator return an error message? b. Is \(\theta=\frac{\pi}{2}\) a solution to the equation? Explain.

Step-by-Step Solution

Answer

Calculator errors at \( \theta = \frac{\pi}{2} \) due to undefined \( \tan \theta \), but analytically \( \theta = \frac{\pi}{2} \) satisfies the equation.

Step 1: Analyzing the Equation

The given equation is \( \cot \theta = \sin \left(\theta - \frac{\pi}{2}\right) \). We need to explore both sides of this equation to understand its behavior within the specified interval \( 0 \leq \theta \leq \pi \).

Step 2: Understand \( \cot \theta \) and \( \sin \left(\theta - \frac{\pi}{2}\right) \)

\( \cot \theta = \frac{1}{\tan \theta} \) and \( \sin \left(\theta - \frac{\pi}{2}\right) = -\cos \theta \), due to the phase shift property of the sine function. Thus, the equation can be rewritten as \( \frac{1}{\tan \theta} = -\cos \theta \).

Step 3: Recognize Values that Result in Undefined Expressions

The expression \( \cot \theta = \frac{1}{\tan \theta} \) becomes undefined at \( \theta = \frac{\pi}{2} \), because \( \tan \left(\frac{\pi}{2}\right) \) is undefined.

Step 4: Understanding Calculator Behavior

The calculator returns an error for \( \theta = \frac{\pi}{2} \) because \( \tan \theta \) is undefined at this point, leading to \( \cot \theta \) being undefined. This means there is no value to compute at \( \theta = \frac{\pi}{2} \).

Step 5: Verifying \( \theta = \frac{\pi}{2} \) as a Solution

Despite the calculator error, substituting \( \theta = \frac{\pi}{2} \) into the transformed equation \( \cot \theta = -\cos \theta \) results in both sides being equal to zero when excluding the undefined aspect of \( \cot \theta \). Thus, it is a solution from an analytical perspective.

Key Concepts

Cotangent Function

The cotangent function, often symbolized as \( \cot \theta \), is a trigonometric function that is defined as the reciprocal of the tangent function. In mathematical terms, if \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), then \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). This function is particularly interesting because its value is highly dependent on the angles for which it is defined. When \( \theta = \frac{\pi}{2} \), the value of \( \tan \theta \) becomes undefined, as the cosine of this angle is zero. Therefore, \( \cot \theta \), being the reciprocal, is also undefined at these points.

Key points to remember about the cotangent function:

• It is undefined wherever \( \tan \theta \) is zero, such as at odd multiples of \( \frac{\pi}{2} \).
• The value can be positive or negative, depending on the quadrant where \( \theta \) resides.
• Understanding its behavior is crucial to solving trigonometric equations involving \( \cot \theta \).

Sine Function

The sine function, represented as \( \sin \theta \), is one of the fundamental trigonometric functions that define the relationship between an angle and the lengths of the sides in a right triangle. The range of the sine function is from -1 to 1, and it is known for its smooth, periodic wave-like graph.

The sine of an angle \( \theta \) can be visualized in the unit circle as the y-coordinate. As \( \theta \) increases, the sine function exhibits a repeating pattern every \( 2\pi \) radians. Key characteristics include:

• Periodic with a period of \( 2\pi \).
• Range between -1 and 1.
• Symmetrical about the origin, indicating it is an odd function.

When considering the expression \( \sin(\theta - \frac{\pi}{2}) \), this is a compelling concept because it introduces a phase shift, altering how the function is perceived on the graph.

Phase Shift

A phase shift refers to the horizontal movement of a trigonometric function's graph on the coordinate plane. When we consider expressions like \( \sin(\theta - \frac{\pi}{2}) \), the sine curve is shifted horizontally by \( \frac{\pi}{2} \) units.

This phase shift is crucial in altering the starting position of the wave pattern:

• A negative shift, such as \( \theta - \frac{\pi}{2} \), moves the function right.
• A positive shift, say \( \theta + \frac{\pi}{2} \), moves the function left.
• It does not affect the amplitude; only the function's starting point along the x-axis.

Understanding phase shifts is essential, especially when graphing functions and predicting how different transformations will affect outcomes in trigonometric equations.

Undefined Expressions

Undefined expressions in trigonometry arise when we encounter values that result in infinite or non-real outputs. Specifically, functions like \( \tan \theta \) and \( \cot \theta \) become undefined at certain critical angles.

For instance, \( \tan \theta \) is undefined at \( \theta = \frac{\pi}{2} + k\pi \) where \( k \) is an integer because the cosine component of the tangent formula is zero and arithmetic with zero in the denominator is undefined.

• In our example, \( \cot \theta = \frac{1}{\tan \theta} \) becomes undefined whenever \( \tan \theta \) is.
• This happens predominantly at odd multiples \( \frac{\pi}{2} \).

Recognizing when these undefined expressions occur is key to solving trigonometric equations, as it affects the validity of the solutions within certain intervals.