Problem 16 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin x-\sin 2 x $$

Step-by-Step Solution

Answer

The expression \( \sin x-\sin 2 x \) can be simplified and expressed as a product \( \sin x (1 - 2 \cos x) \)

Step 1: Identify Relevant Trigonometric Identities

To approach this problem, make use of the double angle identity for sine, which is \( \sin 2x=2 \sin x \cos x \)

Step 2: Substituting the Identity

Insert this identity into the original expression, replacing \( \sin 2x \) with \( 2 \sin x \cos x \) to get the new expression to be \( \sin x - 2 \sin x \cos x \)

Step 3: Factor out Common Factor

Now, there's a common factor of \( \sin x \) in each term. Factor this out to get \( \sin x (1 - 2 \cos x) \)

Key Concepts

Double Angle Identity

In trigonometry, identities are equations that are true for all values of the involved variables. The double angle identity is one such identity that provides a way to relate angles in trigonometric functions that are twice as large. For the sine function, the double angle identity is: \[ \sin 2x = 2 \sin x \cos x \]This identity expresses the sine of a doubled angle, \(2x\), in terms of the sine and cosine of the original angle, \(x\). This expression is quite useful when simplifying trigonometric expressions that involve products or sums of angles. When tackling a problem like \( \sin x - \sin 2x \), knowing the double angle identity allows you to simplify \( \sin 2x \) into a more nearly related function of \( x \), which can lead the way to a clearer and more manageable expression. Using these identities wisely can often make seemingly complex trigonometric problems much easier to solve.

Factoring Expressions

Factoring is a core algebraic process used to simplify expressions or solve equations, and it's crucial when working with trigonometric identities. In the context of the trigonometric expression \( \sin x - 2 \sin x \cos x \), factoring involves finding a common factor that can be extracted to simplify the expression.

• Identify any common factors in the terms. In our case, \( \sin x \) is a common factor.
• Extract the common factor from each term. For \( \sin x - 2 \sin x \cos x \), factoring out \( \sin x \) gives us: \[ \sin x (1 - 2 \cos x) \]

Factoring not only simplifies expressions but also helps in visualizing the relationship between trigonometric terms. This facilitates finding the product of expressions and can also be useful for solving trigonometric equations.

Sine Function

The sine function is a fundamental component in trigonometry and is defined for angles in a right triangle as the ratio of the length of the opposite side to the hypotenuse. Mathematically, if \( \theta \) is an angle, then:\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]Beyond triangles, the sine function extends across the entire field of trigonometry and calculus as a periodic function. One full cycle of the sine wave is completed over \(2\pi\) radians or 360 degrees, with its values oscillating smoothly between -1 and 1.Some key properties of the sine function include:- **Periodicity:** The sine function repeats its values in a regular cycle, making it periodic.- **Odd Function:** It satisfies \( \sin(-x) = -\sin x \). This means it is symmetric with respect to the origin.Understanding the sine function is essential when dealing with any trigonometric identity or expression, as many complex identities boil down to basic sine or cosine values, altering perceptions of more difficult problems by connecting them with familiar trigonometric functions.