Question

Exploration Let \(y=a \sin x+b \cos x\) Use the symbolic manipulator of a computer algebra system (CAS) to help you with the following: (a) Express y as a sinusoid for the following pairs of values: a=2, b=1 ; \quad a=1, b=2 ; \quad a=5, b=2 ; \quad a=2, b=5 a=3, b=4 (b) Conjecture another formula for \(y\) for any pair of positive integers. Try other values if necessary. (c) Check your conjecture with a CAS. (d) Use the following formulas for the sine or cosine of a sum or difference of two angles to confirm your conjecture. \(\begin{aligned} \sin \alpha \cos \beta & \pm \cos \alpha \sin \beta=\sin (\alpha \pm \beta) \\ \cos \alpha \cos \beta \pm \sin \alpha \sin \beta &=\cos (\alpha \mp \beta) \end{aligned}\)

Short answer

The conjectured formula for \(y\) for any pair of positive integers is \( y = \sqrt{{a}^2 + {b}^2} * \sin\left(x + \arctan\left(\frac {b}{a}\right)\right)\), which has been confirmed through comparisons with a Computer Algebra System and via a trigonometric identity. This formula also proves to be equivalent to the initial expressions for the given pairs of 'a' and 'b'.

Step-by-step solution

Express \(y\) as a Sinusoid

Use the formula \(y = R * \sin(x + \alpha)\), where \( R = \sqrt {{a}^2 + {b}^2}\) and \(\alpha = \arctan(\frac{b}{a})\). These calculations will provide the equivalent sinusoidal expressions for different 'a' and 'b' values provided. For instance, when a=2, b=1, the required values would be \(R= \sqrt{5}, \alpha = \arctan(\frac{1}{2})\). Do the same for other given pairs.

Conjecture a formula for \(y\)

The formulas for 'y' for each pair of 'a' and 'b' seem to share a common factor. It might be observed that for any pair of positive integers, we can come up with a general formula : \( y = \sqrt{{a}^2 + {b}^2} * \sin\left(x + \arctan\left(\frac {b}{a}\right)\right)\)

Check the Conjecture with a CAS

Substitute some different pairs of positive integers into the conjectured formula and compare the results with a Computer Algebra System. They should coincide, confirming the conjecture's validity.

Confirm the Conjecture by Trigonometry

To confirm this conjecture, make use of the identity \(\sin(\alpha + \beta) = \sin\alpha * \cos\beta + \cos\alpha * \sin\beta\). Substitute \(\alpha = x\) and \(\beta = \arctan(\frac {b}{a})\) into this identity. Considering \(\tan\beta = \frac {b}{a}\), we see that \(\cos\beta = \frac {a}{\sqrt{{a}^2 + {b}^2}}\) and \(\sin\beta = \frac {b}{\sqrt{{a}^2 + {b}^2}}\). This will return the original expression \( a * \sin x + b * \cos x\), hence confirming the conjecture.