Problem 27 - Algebra and Trigonometry

Question

Prove the identity. $$ \sin (x-\pi)=-\sin x $$

Step-by-Step Solution

Answer

The identity \( \sin(x - \pi) = -\sin x \) is proven using the subtraction formula.

Step 1: Apply the Sine Subtraction Identity

Recall the sine subtraction identity, which states: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]Using this identity, substitute \(a = x\) and \(b = \pi\), resulting in:\[ \sin(x - \pi) = \sin x \cos \pi - \cos x \sin \pi \]

Step 2: Evaluate Trigonometric Functions

Now, substitute the values for \( \cos \pi \) and \( \sin \pi \). We know that \( \cos \pi = -1\) and \( \sin \pi = 0\). Thus:\[ \sin(x - \pi) = \sin x (-1) - \cos x (0) \]

Step 3: Simplify the Expression

Simplify the expression obtained in the previous step:\[ \sin(x - \pi) = -\sin x - 0 \cdot \cos x \]This simplifies to:\[ \sin(x - \pi) = -\sin x \]

Step 4: State the Conclusion

Based on the simplification, we have shown that:\[ \sin(x - \pi) = -\sin x \]Thus, the identity is proven as required.

Key Concepts

Sine Subtraction Formula

The sine subtraction formula is a fundamental identity in trigonometry, which helps to express the sine of the difference between two angles. This formula is given by:

• \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)

It is derived from the basic definition of the sine and cosine functions in the unit circle.
To use this formula efficiently, substitute the angles for \(a\) and \(b\) into the formula and apply known values or properties of the sine and cosine functions.
For example, in the identity \( \sin(x - \pi) = -\sin x \), we set \(a = x\) and \(b = \pi\). By knowing the standard trigonometric values \( \cos \pi = -1\) and \( \sin \pi = 0\), we can utilize the formula to verify trigonometric identities.

Proofs in Trigonometry

Trigonometric proofs are essential for understanding relationships between different trigonometric functions and identities.
A trigonometric proof involves starting with one side of an equation and using known identities and properties to transform it into the other side.

• This is done by substituting values, rearranging terms, and simplifying expressions.

In the given problem, we start with \(\sin(x - \pi)\) and use the sine subtraction identity, then substitute known values for \(\cos \pi\) and \(\sin \pi\).
After simplifying, we demonstrate the equation simplifies to \(-\sin x\), hence proving the identity. These steps not only verify mathematical truths but also enhance comprehension of how cosine and sine function properties relate to each other.

Trigonometric Functions Evaluation

Evaluating trigonometric functions at specific angles helps us solve equations and verify identities.
Knowing key values, like \( \cos(0) = 1\), \( \cos(\pi) = -1\), \( \sin(0) = 0\), and \( \sin(\pi) = 0\), is crucial for these calculations.

• When substituting these in formulae, they simplify the expressions greatly.
• Example: In \(\sin(x - \pi) = \sin x \cos \pi - \cos x \sin \pi\), substitute \(\cos \pi = -1\) and \(\sin \pi = 0\).

This results in \(\sin(x - \pi) = -\sin x - 0\cdot \cos x\), simplifying further to \(-\sin x\).
Understanding how to evaluate trigonometric functions is fundamental for simplifying expressions and discovering insights into periodicity and symmetry in trigonometry.