Problem 135 - Precalculus with Limits

Question

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer. \( \dfrac{\left(\sin k\theta\right)}{\left(\cos k\theta\right)} = \tan \theta \), \( k \) is a constant.

Step-by-Step Solution

Answer

The given equation is an identity only if the constant k is equal to 1.

Step 1: Understand the given equation

We have this given equation \(\frac{\sin k\theta}{\cos k\theta} = \tan \theta\). Now we need to understand what is asking. We are asked to prove if the equation is an identity. The terms \(\sin k\theta\) and \(\cos k\theta\) are in the form of sine and cosine functions respectively and \(\tan \theta\) is a tangent function.

Step 2: Application of trigonometric property

The property of tangent \(\tan \theta\) is equal to \(\frac{sin \theta}{cos \theta}\). By using this property, our equation transforms into \(\frac{\sin k\theta}{\cos k\theta}\) is equal to \(\frac{\sin \theta}{\cos \theta}\) from the given equation \(\frac{\sin k\theta}{\cos k\theta} = \tan \theta\).

Step 3: Compare and analyze

With the transformation, we have got two expressions, we can compare these expressions as follows: \(sin k\theta = sin \theta\) and \(cos k\theta = cos \theta\). This implies that \(k\theta = \theta\), which will be true only when k=1.

Step 4: Final evaluation

From the comparison of the transformed equation, we can state that: The given equation \(\frac{\sin k\theta}{\cos k\theta} = \tan \theta\) is an identity only when the constant k=1.

Key Concepts

Understanding the Sine Function

The sine function, often written as \( \sin \theta \), is one of the primary trigonometric functions. It applies to angles in right triangles and represents the ratio of the opposite side to the hypotenuse. Understanding this function is essential:

• For a right triangle, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• On the unit circle, it represents the y-coordinate of a point formed by an angle \( \theta \) from the positive x-axis.

In this exercise, \( \sin k\theta \) means that the angle has been modified by a constant \( k \). This can stretch or compress the angle, affecting the value of the sine function.
To determine if the given equation is an identity, it's crucial to see if changing \( k \) alters the equation in a way that breaks the identity. The sine function's periodic and oscillating nature means that changes can have significant effects.

Understanding the Cosine Function

The cosine function, denoted as \( \cos \theta \), complements the sine function. It describes the ratio of the adjacent side to the hypotenuse in a right triangle:

• For a right triangle, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• On the unit circle, it corresponds to the x-coordinate of a point determined by angle \( \theta \) from the origin.

The function \( \cos k\theta \) behaves similarly to sine when the angle undergoes modification by the constant \( k \).
In the context of proving or disproving the equation as an identity, it's about seeing how these modifications affect both sides of the identity. If \( \cos k\theta \) equals \( \cos \theta \), then it implies no change in angle, necessitating that \( k = 1 \), confirming the proposition under specific conditions.

Understanding the Tangent Function

The tangent function, represented as \( \tan \theta \), is another fundamental trigonometric function. It is the ratio of the sine function to the cosine function:

• Mathematically, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Visually, it represents the slope of the line formed from the origin through a point on the unit circle.

In this exercise, understanding that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is crucial.
The problem requires establishing if \( \frac{\sin k\theta}{\cos k\theta} \) can equal \( \tan \theta \) universally. By manipulating the trigonometric properties, you recognize this equality would only hold if \( k = 1 \), as modifying \( \theta \) with a factor changes the resulting tangent value, disrupting the identity unless properly aligned.