Question

Describe and correct the error in simplifying the expression. $$ \begin{aligned} \sin \left(x-\frac{\pi}{4}\right) &=\sin \frac{\pi}{4} \cos x-\cos \frac{\pi}{4} \sin x \\ &=\frac{\sqrt{2}}{2} \cos x-\frac{\sqrt{2}}{2} \sin x \\ &=\frac{\sqrt{2}}{2}(\cos x-\sin x) \end{aligned} $$

Short answer

The error in simplifying the given expression is in the third stage of the simplification. The third line incorrectly factors \(\sqrt{2}/2\) out of \(\sqrt{2}/2 \cos x - \sqrt{2}/2 \sin x\). The correct expression is \(\sqrt{2}/2 \cos x - \sqrt{2}/2 \sin x\), meaning that each term should be considered separately due to the fact that \(\cos x\) and \(\sin x\) are not identical terms.

Step-by-step solution

Verify the Identity Used

The first step is to confirm that the identity used in the problem is correct. Remembering the identity for the sine of a difference, \(\sin(a - b) = \sin a \cos b - \cos a \sin b\), we notice that the identity is correctly applied in the first step of the given work.

Evaluate the Sine and Cosine of \(\pi/4\)

The second step is to evaluate the sine and cosine of \(\pi/4\), which are both \(\sqrt{2}/2\). In the given work, these values are correctly used in the second step where \(\sin(\pi/4) \cos x - \sin x \cos(\pi/4)\) is simplified to \(\sqrt{2}/2 \cos x - \sqrt{2}/2 \sin x\).

Spot the Error

In Step 3, an error is made when the two terms \(\sqrt{2}/2 \cos x\) and \(-\sqrt{2}/2 \sin x\) are factored into \(\sqrt{2}/2 (\cos x - \sin x)\). However, the original expression cannot be factored in this way, because while the coefficients of \(\cos x\) and \(\sin x\) are the same, \(\cos x\) and \(\sin x\) are not identical terms, hence they should remain separated. The correct simplification should be \(\sqrt{2}/2 \cos x - \sqrt{2}/2 \sin x\).

Key concepts

Sine of a Difference Identity

When you're working with trigonometric expressions, one of the most useful tools in your toolbox is the **sine of a difference identity**. This identity states that for any two angles, \( a \) and \( b \), the expression \( \sin(a - b) \) can be rewritten as \( \sin a \cos b - \cos a \sin b \). This identity is handy because it allows you to break down a complex trigonometric expression into simpler components.

• **Understanding Components**: The formula relies on knowing the values of \( \sin a \), \( \cos a \), \( \sin b \), and \( \cos b \). In many cases, these values are computed directly through known angles such as \( \pi/4 \).
• **Example Application**: In the original exercise, \( a = x \) and \( b = \pi/4 \). Using the identity results in \( \sin(x - \pi/4) = \sin x \cos(\pi/4) - \cos x \sin(\pi/4) \). The student would then substitute the known values of \( \cos(\pi/4) \) and \( \sin(\pi/4) \).

The importance of this identity cannot be overstated, especially when simplifying expressions and solving trigonometric equations where difference angles are involved.

Factoring Expressions

Factoring is a method that can be used to simplify expressions or solve equations by writing them as a product of simpler factors. But not all expressions can be conveniently factored, especially when you're dealing with different trigonometric functions like sine and cosine.

• **Factoring Mistakes**: A common error in trigonometry, as highlighted in the original exercise, involves incorrectly factoring expressions that include terms like \( \cos x \) and \( \sin x \). Even if these terms have common numeric coefficients, they represent different functions and cannot just be factored into a common parenthesis. The original problem attempted to factor \( \sqrt{2}/2 \cos x - \sqrt{2}/2 \sin x \) into \( \sqrt{2}/2 (\cos x - \sin x) \), which is incorrect.
• **Correct Application**: Correct factoring should consider the functions themselves. If the terms are different trigonometric functions, usually, they can only be expressed separately, unless a valid identity allows further simplification.

Always check the nature of the terms involved before applying any factoring strategy.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions means making them as concise and easy to manage as possible. This often involves using identities, known values, or algebraic techniques.

• **Using Identities**: Applying identities like the sine and cosine sum and difference identities can simplify complex angles into manageable forms. This involves turning an expression like \( \sin(x - y) \) into terms that add or subtract known values of sine and cosine.
• **Recognizing Common Values**: Many trigonometric values are standardized; for example, \( \sin(\pi/4) = \cos(\pi/4) = \sqrt{2}/2 \). Knowing these values by heart can speed up the simplification process.
• **Avoid Over-Factoring**: As demonstrated in the original problem, inserting unnecessary factors can lead to errors. It’s often better to verify each step, ensuring that the structure of the trigonometric functions remains intact.

With practice, recognizing which identities to use and when to use them will become second nature, making simplification easier and more intuitive.