Problem 133 - Precalculus

Question

For the following exercises, prove the identity given. $$ \sin (2 x)=-2 \sin (-x) \cos (-x) $$

Step-by-Step Solution

Answer

Identity is proven as both expressions simplify to \( 2 \sin(x) \cos(x) \).

Step 1: Understanding the identity

We need to show that the left side of the equation, \( \sin(2x) \), equals the right side, \(-2 \sin(-x) \cos(-x)\). To do this, we'll rewrite both sides using known trigonometric identities.

Step 2: Use sin(double angle) identity

The double angle identity for sine is \( \sin(2x) = 2 \sin(x) \cos(x) \). We'll use this form to express the left side of the identity.

Step 3: Simplify the right side using sine and cosine properties

Use the odd and even properties of sine and cosine: \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \). Thus, \(-2\sin(-x)\cos(-x) = -2(-\sin(x))\cos(x) = 2\sin(x)\cos(x)\).

Step 4: Compare both expressions

Now, both sides are expressed as \( 2 \sin(x) \cos(x) \). Since they are equal, the original identity \( \sin(2x) = -2 \sin(-x) \cos(-x) \) is proven.

Key Concepts

Double Angle Identity

The double angle identity is a crucial concept in trigonometry that helps to simplify expressions and solve various trigonometric equations. It specifically deals with finding trigonometric functions of double angles, such as \(2x\). The identity for sine in this context is:

• \( \sin(2x) = 2\sin(x)\cos(x) \)

This identity allows us to express the sine of a double angle using both the sine and cosine of the original angle, \(x\). By understanding and applying this formula, complex trigonometric expressions involving double angles become more manageable. It is especially useful in proving identities and solving equations where double angles appear.

Odd and Even Functions

In trigonometry, functions are often classified as odd or even. Understanding whether a trigonometric function is odd or even helps in simplifying expressions, as seen in the original exercise. A function \(f(x)\) is odd if \(f(-x) = -f(x)\). Conversely, a function is even if \(f(-x) = f(x)\).

• The sine function is odd: \( \sin(-x) = -\sin(x) \)
• The cosine function is even: \( \cos(-x) = \cos(x) \)

This characteristic is essential for manipulating trigonometric expressions, as it helps to replace negative angles and further simplify or prove identities like in the given exercise.

Sine Function

The sine function is one of the fundamental trigonometric functions. It relates the ratios of sides in a right triangle to its angles. For an angle \(x\), the sine function is denoted as \(\sin(x)\).

• The sine function has a range of values between -1 and 1.
• It is periodic with a period of \(2\pi\), repeating its values every \(2\pi\) radians or 360 degrees.

Understanding sine's nature is crucial for working with trigonometric identities, modeling wave forms, and much more. The double angle identity incorporates sine, emphasizing its role in constructing and simplifying expressions.

Cosine Function

The cosine function complements the sine function as another primary trigonometric function. When dealing with angles in a right triangle, the cosine of an angle \(x\) is defined as \(\cos(x)\).

• The cosine function ranges from -1 to 1 along its domain.
• Similar to sine, it also has a periodic cycle of \(2\pi\).

Cosine's even nature, where \(\cos(-x) = \cos(x)\), significantly simplifies calculations in trigonometric identities. Knowing how both sine and cosine work together makes proving and solving equations, like the one in the exercise, straightforward.