Problem 26 - Precalculus Essentials

Question

Verify each identity. $$\frac{\cos 4 x-\cos 2 x}{\sin 2 x-\sin 4 x}=\tan 3 x$$

Step-by-Step Solution

Answer

The given trigonometric identity \(\frac{cos(4x) - cos(2x)}{sin(2x) - sin(4x)} = tan(3x)\) is verified successfully.

Step 1: Write both sides in terms of sine and cosine

The given identity is already written in terms of sine and cosine, so there's no need to do anything in this step for this particular problem.

Step 2: Express 4x in terms of 2x

Using the formula of double angle, \(cos(4x) = 1 - 2sin²(2x)\) and \(sin(4x) = 2sin(2x)cos(2x)\). Substitute these in the main equation to simplify it.

Step 3: Simplify the equation

After substitution, we get \(\frac{1 - 2sin²(2x) - cos(2x)}{2sin(2x)cos(2x) - sin(2x)}\). We can express cos(2x) = 1 - 2sin²(x) in this equation. And then we see that 2sin²(2x) eventually cancels out from the numerator and denominator. Now, we get \(\frac{1 - cos(2x)}{2sin(2x)cos(2x) - sin(2x)} = \frac{2sin²(x)}{2sin(x)cos(x)}\). We can cancel out the common factors.

Step 4: Get the final verification

After cancelling out the common factors, we left with \(tan(3x)\) which is the right side of the given identity. So, we have verified the given trigonometric identity successfully.

Key Concepts

Trigonometric Double Angle Formulas

Trigonometric double angle formulas are a cornerstone of simplifying complex trigonometric expressions. They describe the sine, cosine, and tangent of an angle that is double another angle in terms of the trigonometric functions of the original angle.
For instance, the double angle formulas for sine and cosine are given by:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta) \)
• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

These formulas are incredibly useful in solving trigonometric equations and in verifying identities, like turning \( \cos(4x) \) into \( 1 - 2\sin^2(2x) \) and \( \sin(4x) \) into \( 2\sin(2x)\cos(2x) \), as seen in the original exercise. Without these, simplifying such an equation would be much more complicated, if not impossible.

Sine and Cosine

Sine and cosine are fundamental to trigonometry. The sine of an angle in a right triangle describes the ratio of the length of the opposite side to the length of the hypotenuse, while cosine gives the ratio of the adjacent side to the hypotenuse.

Unit Circle Representation

If we take the unit circle, sine and cosine represent the y and x coordinates, respectively, of a point on the circle corresponding to an angle \( \theta \) from the positive x-axis. Their importance is further highlighted by their role in the Pythagorean identity, \( \sin^2(\theta) + \cos^2(\theta) = 1 \), which itself is a backbone of trigonometric identities. Understanding how to manipulate expressions involving sine and cosine is essential for verifying trigonometric identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a practical skill in mathematics, especially when dealing with complex equations or verifying identities. This process often involves several steps:

• Identifying and applying appropriate trigonometric identities
• Factoring and canceling common terms
• Converting complex terms into simpler ones

The original exercise showcases this process by transforming the given expression into a simpler form that reveals the underlying identity. Various identities, such as the double angle formula, are employed strategically to break down the complex numerator and denominator, eventually simplifying down to the tangent function. Simplification can sometimes reveal that terms in the numerator and the denominator can be canceled out, as seen with the common factor of \( \sin(2x) \) in the exercise.

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is another primary trigonometric function and is defined as the ratio of the sine and cosine of an angle \( \theta \), that is, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Properties of Tangent

Some key characteristics include its periodicity with a period of \( \pi \) radians and its undefined value when the cosine of an angle is zero, as this would mean division by zero. The original problem ends with the tangent of three times the angle x, and this simplification was crucial in confirming the identity by demonstrating that both sides of the equation are equivalent. This use of the tangent function highlights its relation to sine and cosine and its relevance in transforming and simplifying trigonometric identities.