Question

Exploration Let y=\sin (a x)+\cos (a x) Use the symbolic manipulator of a computer algebra system (CAS) to help you with the following: (a) Express \(y\) as a sinusoid for \(a=2,3,4,\) and 5 (b) Conjecture another formula for \(y\) for \(a\) equal to any positive integer \(n .\) (c) Check your conjecture with a CAS. (d) Use the formula for the sine of the sum of two angles (see Exercise 56 \(\mathrm{c}\) ) to confirm your conjecture.

Short answer

The transformed sinusoid given the equation is \(y = \sqrt{2n^{2}}\sin(x + \frac{\pi}{4})\) for an \(n\) positive integer, which was confirmed via the sine of the sum of two angles formula.

Step-by-step solution

Transformation into sinusoidal expression

Starting with the equation \(y=\sin (a x)+\cos (a x)\), we use the general formula for the sum of sine and cosine functions, which is \(R\sin(x + α)\), where \(R = \sqrt{a^{2}+b^{2}}\) is the amplitude, and \(α = \arctan(\frac{b}{a})\) is the phase shift. Using this and a computer algebra system (CAS) to manipulate the equation, the equation can be expressed as a sinusoid. For \(a = 2, 3, 4, 5\), the formulas would respectively be: \(y_{2} = 2\sqrt{2}\sin(x + \frac{\pi}{4})\), \(y_{3} = 2\sqrt{2}\sin(x + \frac{\pi}{4})\), \(y_{4} = 2\sqrt{2}\sin(x + \frac{\pi}{4})\), \(y_{5} = 2\sqrt{2}\sin(x + \frac{\pi}{4})\)

Conjecture a formula

Putting \(a = n\), where \(n\) is any positive integer in the above formula, we should get that \(y_{n} = \sqrt{2n^{2}}\sin(x + \frac{\pi}{4})\)

Validating Conjecture using CAS

Use a computer algebra system (CAS) to input and verify the formula determined above for \(n\) being a positive integer. If the output matches the conjecture, then it's correct.

Confirming conjecture using sine sum formula

The formula for the sine of the sum of two angles can be stated as \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\). This can be used to prove the validity of the conjecture as \(\sin(ax + \frac{\pi}{4}) = \sin(ax)\cos(\frac{\pi}{4}) + \cos(ax)\sin(\frac{\pi}{4})\). This matches with the original equation (\(y=\sin (a x)+\cos (a x)\)) which verifies our conjecture finally.

Key concepts

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is an invaluable tool for mathematics students, especially when dealing with complex expressions such as sinusoidal functions. It is a software program capable of performing symbolic mathematics. Put simply, a CAS can manipulate mathematical expressions, solve equations, and perform algebraic operations with variables, as opposed to just numbers.
For example, when given the function y = \(sin(ax) + cos(ax)\), a CAS can symbolically transform this expression into a single sinusoidal form. This not only streamlines the solving process but also aids in visualizing and understanding the behavior of trigonometric functions. A CAS becomes your digital math companion, proving or rejecting conjectures by applying extensive computational rules that exceed manual calculations.

Transformation of Trigonometric Functions

The transformation of trigonometric functions into a sinusoidal form is a pivotal skill in calculus. This involves representing a combination of sine and cosine functions as a single sine (or cosine) function. The advantage of this transformation is that it simplifies the original expression and can reveal important characteristics such as amplitude and phase shift.

Why Transform?

• To identify the function's amplitude and phase shift easier,
• To simplify the sum of sine and cosine functions into a more manageable form,
• To prepare the function for integration or differentiation with less complexity.

By using the relationship R\sin(x + \alpha), where R and \(\alpha\) respectively represent the amplitude and phase shift, one expresses the original function in a format more suitable for further analysis or computation.

Sine of the Sum of Two Angles

This fundamental concept in trigonometry relates to how one can express the sine of a sum of two angles, say \(\alpha + \beta\), as a combination of the sines and cosines of the separate angles: \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\). This identity is instrumental in tasks such as transforming trigonometric expressions and solving trigonometric equations.
It allows one to link and transform one type of trigonometric function into another, paving the way to unify the given functions into a singular, sinusoidal expression. This concept is essential when verifying conjectures related to the sum of angles, as in the case of combining sinusoidal functions like sine and cosine with the same argument.

Conjecture Verification

Conjecture verification is the process of testing a mathematically informed guess or hypothesis, usually based on patterns or observations, with rigorous methods. In the context of the given exercise, if one conjectures a formula for the sum of sine and cosine with a variable coefficient, a CAS can be deployed to check this conjecture.
A conjecture might arise from noticing patterns or through algebraic manipulations. Once a pattern is noticed, for instance, that a function behaves in a predictable way for several integer values, one might suggest a general rule that applies to all integers. The CAS's role then becomes checking whether this is true by plugging in numerous values and ensuring the conjectured formula holds, which adds weight to the initial supposition. This sort of computational confirmation is an essential step in mathematics before a conjecture can be considered viable or before trying to prove it with formal mathematical proofs.