Evaluate sin(pi/4)-sin(pi/4)

Solution:

Step 1: Break down each term.

• Step 1.1: Determine the precise value of $sin\left(\frac{\pi}{4}\right)$, which is $\frac{\sqrt{2}}{2}$. Now express the equation as $\frac{\sqrt{2}}{2} - sin\left(\frac{\pi}{4}\right)$.

• Step 1.2: Again, the precise value of $sin\left(\frac{\pi}{4}\right)$ is $\frac{\sqrt{2}}{2}$. Replace the sine term to get $\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$.

Step 2: Combine and simplify the expression.

• Step 2.1: Use a common denominator to merge the terms, resulting in $\frac{\sqrt{2} - \sqrt{2}}{2}$.

• Step 2.2: Subtract $\sqrt{2}$ from $\sqrt{2}$ to get $\frac{0}{2}$.

• Step 2.3: Simplify the fraction by dividing $0$ by $2$ to obtain the final result of $0$.

Knowledge Notes:

The problem involves simplifying a trigonometric expression by using known values of sine at specific angles. The steps taken to solve the problem are as follows:

1. Trigonometric Functions: Understanding that trigonometric functions like sine, cosine, and tangent have specific values at certain standard angles (like $\frac{\pi}{4}$, $\frac{\pi}{2}$, etc.) is crucial. For $\frac{\pi}{4}$, the sine value is known to be $\frac{\sqrt{2}}{2}$.

2. Simplification: The process involves simplifying the expression by substituting the exact values of the trigonometric functions and then performing arithmetic operations like subtraction and division.

3. Arithmetic Operations: The steps include combining like terms, which in this case are the same trigonometric values, and then subtracting them. When subtracting identical numbers, the result is zero.

4. Zero Property: Dividing zero by any non-zero number results in zero, which is the final step in the problem.

5. LaTeX Formatting: The solution uses LaTeX to properly format mathematical expressions, ensuring clarity and precision in presenting the problem and solution.

By understanding these concepts, one can effectively solve similar problems involving trigonometric expressions and their simplification.