Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

For the trigonometrically strong. Show that $$ A_{t}=\rho^{t} \sin t \theta \quad \text { solves } \quad A_{t+2}-p A_{t+1}+q A_{t}=0 $$ where $$ p^{2}-4 q<0, \quad \rho^{2}=q \quad \text { and } \quad \theta=\arccos \left(\frac{p}{2 \rho}\right) $$ You may wish to know that \(\sin (x+y)+\sin (x-y)=2 \sin x \cos y\)

Step-by-Step Solution

Answer

The series \( A_t = \rho^t \sin t \theta \) satisfies the recurrence because \( p^2 - 4q < 0 \), \( \rho^2 = q \), and trigonometric identities allow simplification to zero.

Step 1: Identify Known Variables and Conditions

We are provided with the solution in the form \( A_t = \rho^t \sin t \theta \) and need to show that this satisfies the given recurrence relation \( A_{t+2} - p A_{t+1} + q A_t = 0 \). The conditions are \( p^2 - 4q < 0 \), \( \rho^2 = q \), and \( \theta = \arccos \left( \frac{p}{2\rho} \right) \).

Step 2: Compute Terms of Recurrence

Identify the terms in the recurrence: namely, \( A_{t+1} = \rho^{t+1}\sin((t+1)\theta) \) and \( A_{t+2} = \rho^{t+2}\sin((t+2)\theta) \).

Step 3: Substitute Terms into Recurrence

Substitute the terms \( A_t, A_{t+1}, \text{ and } A_{t+2} \) into the recurrence: \[ \rho^{t+2} \sin((t+2)\theta) - p \rho^{t+1} \sin((t+1)\theta) + q \rho^t \sin(t\theta) = 0. \]Since \( q = \rho^2 \), substitute further to get terms in powers of \( \rho^t \).

Step 4: Apply Sine Addition Formula

Use the identity \( \sin((t+2)\theta) = \sin(t\theta + 2\theta) \) and \( \sin((t+1)\theta) = \sin(t\theta + \theta) \). Recall the identity: \[ \sin(x+y) = \sin x \cos y + \cos x \sin y. \]Apply this to expand both \( \sin(t+1)\theta \) and \( \sin(t+2)\theta \).

Step 5: Use Trigonometric Identities

Owing to the given identity \( \sin(x+y) + \sin(x-y) = 2 \sin x \cos y \), express \( \sin((t+2)\theta) \) and \( \sin((t+1)\theta) \) to simplify:\[ \sin((t+2)\theta) = 2\sin((t+1)\theta)\cos(\theta) - \sin(t\theta). \]Similarly, express \( \sin((t+1)\theta) \) and express terms in terms of \( \sin(\theta) \) and \( \cos(\theta) \).

Step 6: Verify the Recurrence with Trigonometric Conditions

Substitute the expansions back into the recurrence relation and simplify each term using the substitutions made and identities. The conditions \( \cos(\theta) = \frac{p}{2\rho} \) can be used to simplify further. Finally, use \( q = \rho^2 \) to establish the equivalence. Show that each side of the equation simplifies to 0.

Key Concepts

Sine Addition Formula

The sine addition formula is an essential identity in trigonometry, often used to simplify expressions involving angles. The formula is given by:\[\sin(x+y) = \sin x \cos y + \cos x \sin y\]This identity allows us to break down more complex trigonometric expressions into manageable parts by expressing a sine of a sum of angles as a sum of products. In the context of our problem, this helps to express the terms \( \sin((t+1)\theta) \) and \( \sin((t+2)\theta) \) in a way that simplifies incorporating them into a recurrence relation.

We also utilize the identity:\[\sin(x+y) + \sin(x-y) = 2 \sin x \cos y\]This further aids in making expressions involving angle sums more concise. For example, \( \sin((t+2)\theta) \) can be rewritten using \( \sin((t+1)\theta) \) and neutral terms such as \( \cos(\theta) \). This approach makes dealing with trigonometric recurrence relations more straightforward, as it brings us to simpler common forms.

These identities reveal patterns in trigonometric sequences and help to solve trigonometric equations effectively, keeping equations aligned with the given conditions.

Roots of Quadratic Equations

The quadratic equation's roots play a critical role in the analysis of recurrence relations. Consider a quadratic equation:\[ax^2 + bx + c = 0\] Its roots are determined by the discriminant:\(b^2 - 4ac\). In this case, for the stability of the trigonometric recurrence relation given in our problem, we require\( p^2 - 4q < 0\). This condition implies complex roots, which ensure an oscillatory solution when incorporated into the trigonometric framework.

When roots are complex, they can be expressed in a trigonometric form using Euler’s formula, exploiting the periodic nature typical in trigonometry. When we express the solutions as complex numbers, it shows why the expressions are oscillatory - the imaginary part leads to a sine and cosine representation, fitting perfectly in trigonometry.

This means that understanding the nature of roots provides insight into the behavior of solutions to the recurrence relation. Through these complex roots, linked with conditions such as \( \rho^2 = q \), we seamlessly tie together quadratic behavior and trigonometric recurrence.

Complex Numbers in Trigonometry

Complex numbers extend beyond the real number line, providing a robust way to handle periodic phenomena like waves and oscillations. Using Euler's formula, complex numbers tie closely with trigonometric functions:\[e^{ix} = \cos x + i \sin x\]This reveals how complex numbers inherently express sinusoidal functions, which are integral to solving trigonometric recurrence relations. For example, in our context, the expression \( \rho^t \sin t \theta \) is closely tied with complex exponentials due to the periodic, oscillatory characteristic introduced by sine functions.

Connecting complex numbers with trigonometric functions adds a rich algebraic toolkit. It allows converging recurrence characteristics, such as phase shifts or amplitude changes, in a unified manner. With conditions the equation \( p^2 - 4q < 0 \) introduces, we anticipate novel aproaches stemming from complex solutions.The imaginary parts of complex roots don't just provide solutions; they bring a frequency and phase element to repeating patterns seen in trigonometric sequences. Thus, complex numbers capture the essence of trigonometric behavior, explaining persistent cycles within recurrence relations.