Problem 25 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \sin (x-\pi)=-\sin x $$

Step-by-Step Solution

Answer

The identity \( \sin(x-\pi) = -\sin x \) is true, as shown by the sine difference identity and trigonometric evaluations.

Step 1: Recall the Sine Difference Identity

The formula for the sine of a difference is \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). We will use this formula to expand \( \sin(x-\pi) \).

Step 2: Substitute into the Formula

Here, \( a = x \) and \( b = \pi \). Substitute into the identity: \( \sin(x-\pi) = \sin x \cos \pi - \cos x \sin \pi \).

Step 3: Evaluate Trigonometric Functions

We know \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substitute these values into the equation: \( \sin(x-\pi) = \sin x (-1) - \cos x (0) \).

Step 4: Simplify the Expression

Simplify the equation: \( \sin(x-\pi) = -\sin x - 0 \). Therefore, \( \sin(x-\pi) = -\sin x \).

Step 5: Confirm the Identity

We've derived \( \sin(x-\pi) = -\sin x \), confirming that the given identity is indeed true.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It's all about the ratios in a right triangle.
Think of it as measuring how high the triangle rises compared to the longest side.
The sine function, usually written as \(\sin(x)\), deals with angles and circles in a unique way.

• In a right triangle, the sine of an angle \(x\) is the opposite side divided by the hypotenuse.
• On the unit circle, \(\sin(x)\) is the y-coordinate of a point.

The sine function is periodic, meaning it repeats its values in regular intervals. It's connected intricately with other functions like cosine and works via radians and degrees, though radians are often preferred in mathematical contexts.

Difference Identity

The sine difference identity is key when dealing with expressions involving the difference of two angles.
It helps simplify complex trigonometric expressions in a straightforward manner.
Simply put, it's a formula that breaks apart \( \sin(a-b) \).

• The formula is \(\sin(a-b) = \sin a \cos b - \cos a \sin b\).
• We use this formula by substituting specific angle values like \(a = x\) and \(b = \pi\).

This identity shows you how subtracting angles affects the sine, offering flexibility in manipulating trigonometric equations. It particularly helps resolve situations involving angle transformations, providing clarity and simplicity when proving identities.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, used widely in mathematics to solve problems involving angles and distances.
There are three primary functions, sine, cosine, and tangent, each reflecting different ratios in triangles and circles.
Understanding how these functions work simplifies the study of geometry, physics, and engineering.

• **Sine (\(\sin\)** measures the ratio of opposite side over hypotenuse.
• **Cosine (\(\cos\)** represents adjacent side over hypotenuse.
• **Tangent (\(\tan\)** is the opposite over adjacent sides ratio.

These functions are not only essential in solving triangles but also play a role in modeling cyclical phenomena like sound waves and light. Trigonometric identities, such as the one explored here with the difference identity, help in transforming and relating these functions for simpler calculation and understanding.