Problem 24 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

After applying the sum formulas for sine and cosine to the given equation, dividing each term by \(cos(x)\), and simplifying, the left side equation reduces to \(tan(2x)\), verifying the given identity.

Step 1: Apply sum of sines and cosines formulas

Rewrite the expression \(\sin(3x)\) and \(\cos(3x)\) as \(\sin(3x)=3sin(x)-4sin^3(x)\) and \(\cos(3x)=4cos^3(x)-3cos(x)\). This gives: \(\frac{\sin x+3sin(x) - 4( sin^3(x) )}{\cos x+4cos^3(x)- 3cos(x)}\)

Step 2: Simplify the equation

Rewrite the expression and divide each term by \(cos(x)\) it simplifies to \(\frac{1+3tan(x)- 4tan^3(x)}{1+4tan^2(x)- 3}\) where \(tan(x)=sin(x)/cos(x)\)

Step 3: Further Simplify the equation

Simplify the expression: \(\frac{tan(x) (3 - 4tan^2(x) )}{(4tan^2(x) - 2 )}\) and by rebate of \(tan(2x) = \frac{2tan(x)}{1-tan^2(x)}\) it reduces to: \(tan(2x)\).

Key Concepts

Sum of Angles

The sum of angles in trigonometry is a fundamental concept that helps with the manipulation and simplification of expressions involving multiple angles. When dealing with expressions like \(\sin(3x)\) and \(\cos(3x)\), you can use the sum of angles identity to express these in terms of \(\sin(x)\) and \(\cos(x)\). For example, the formulas

• \(\sin(3x) = 3\sin(x) - 4\sin^3(x)\)
• \(\cos(3x) = 4\cos^3(x) - 3\cos(x)\)

help break down complex expressions into simpler ones.
This method enables easier calculations and transformations, which is helpful in verifying identities.
Understanding how to work with sums of angles is crucial for solving a variety of trigonometric problems and ensuring the correctness of solutions. Always remember to check mathematical identities since these transformations can occasionally introduce errors.

Trigonometric Functions

Trigonometric functions, such as sine \(\sin\), cosine \(\cos\), and tangent \(\tan\), are the building blocks of trigonometry. They are used to describe the relationships between the angles and sides of triangles.
Each trigonometric function has specific formulas and identities associated with it. For instance, the tangent function is given by:

• \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)

In the exercise, these identities are used to simplify complex expressions.
By breaking down \(\sin(3x)\) and \(\cos(3x)\) using their respective identities, the exercise demonstrates how to represent trig expressions in terms of basic functions like \(\tan(x)\).
Grasping how these functions interact and transform can deeply enhance your problem-solving skills in trigonometry.

Verifying Identities

Verifying identities is an essential skill in trigonometry that involves proving that two different expressions are equivalent. This process often includes using known trigonometric identities and algebraic manipulation.
In the exercise, the goal is to show that the complex expression simplifies to \(\tan(2x)\). We do this by:

• Applying sum of angles identities:
• Simplifying by substituting and reducing terms
• Comparing with known identities such as \(\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}\)

These steps require a solid understanding of trigonometric properties, as well as careful attention to detail to avoid mistakes.
Establishing equivalence through these methods helps support deeper understanding and mastery of trigonometric principles.