Problem 22 You can use the sum and product ... [FREE SOLUTION] (2024)

Expert verified

The Sine Addition Formula is a foundational tool in trigonometry that allows us to express the sine of a sum of two angles in terms of the sine and cosine of the individual angles. Specifically, the formula states that for any angles A and B, \[\sin(A + B) = \sin A \cos B + \cos A \sin B\].

This formula is extremely useful when dealing with trigonometric equations where the argument of the sine function is a sum or difference of two angles. By using this formula, complex trigonometric expressions can be simplified into more manageable terms involving only the sine or cosine of a single angle. This can be a crucial step when trying to solve trigonometric equations, as it transforms them into a form where other algebraic methods, such as factoring or applying the Zero Product Property, can be used.

In the original exercise, an attempt was made to use this formula to transform \(\sin 3\theta\), which is not directly in the form of \(\sin(A + B)\). However, in some cases, it might be possible to rewrite an expression so that the formula can be applied. Understanding when and how to apply the Sine Addition Formula is a skill that can simplify many trigonometric problems.

The Zero Product Property is an essential element of algebra which states that if the product of two or more factors is zero, then at least one of the factors must be zero. In the context of solving equations, this property is very powerful, as it allows us to break down complex equations into simpler ones.

In our original exercise, once the given trigonometric equation is expressed as a product, the Zero Product Property enables us to set each factor equal to zero and solve for the variable independently. This is effectively used in the later steps of the solution where after factoring out the common \(\sin\theta\), the equation takes the form \(\sin\theta(3\sin^2\theta + 1) = 0\), and we can set \(\sin\theta = 0\) and \(3\sin^2\theta + 1 = 0\) individually.

Understanding the Zero Product Property is crucial in mathematics, especially when solving polynomial equations or trigonometric equations like the one we have. It fundamentally relies on the principle that if a product must be zero, then one or more of its factors must be absent – in other words, holding a value of zero.

The Sine Double-Angle Identity is another powerful tool in trigonometry which expresses the sine of double an angle in terms of the sine and cosine of the original angle. The identity is given by: \[ \sin 2\theta = 2\sin\theta\cos\theta\].

This identity is particularly useful for simplifying the expressions in trigonometric equations that contain multiples of an angle, such as 2\theta or 3\theta. By reducing the complexity of the angle's multiple, the Sine Double-Angle Identity can make the equation much more manageable to solve.

In the problem at hand, the Sine Double-Angle Identity was suggested to transform \(\sin 3\theta\) into an expression involving \(\sin\theta\) and \(\cos\theta\). Although the solution process in the exercise mistakenly took a different route, knowing the correct use of this identity would typically allow for the simplification of terms and easier factoring, demonstrating why familiarity with trigonometric identities like these is invaluable for solving trigonometric equations.