Problem 24 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Done! We were able to verify that the original equation is indeed true. So, \(\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x} = \tan 2 x\).

Step 1: Apply the Trigonometric Formula

We first apply the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) on the right-hand side. So, we need to show that \(\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x} = \frac{\sin 2 x}{\cos 2 x} \)

Step 2: Match the Numerators

Manipulate the numerator of the left-hand side by using the sine double angle identity: \(\sin(2x) = 2\sin(x)\cos(x)\). Notice that we can rewrite \(\sin 3x\) as \(\sin(2x + x)\), which can be transformed to \(2\sin(x)\cos(x) + \sin(x) \). So, the numerator becomes \(\sin x + 2\sin(x)\cos(x) + \sin(x) = 2\sin(x) + 2\sin(x)\cos(x) = 2\sin(x)(1+\cos(x))\)

Step 3: Match the Denominators

Manipulate the denominator of the left-hand side using the cosine double angle identity: \( \cos(2x) = \cos^2(x) - \sin^2(x) \). We can rewrite \(\cos 3x\) as \(\cos(2x + x)\), which can be transformed into \( \cos^2(x) - \sin^2(x) + \cos(x) \). So, the denominator becomes \( \cos x + \cos^2(x) - \sin^2(x) + \cos(x) = 2\cos(x)(1 - \sin(x))\)

Step 4: Simplify the Expression

Now, replace the numerator and the denominator from Steps 2 and 3 into the original left-hand side expression. This becomes: \( \frac{2\sin(x)(1+\cos(x))}{2\cos(x)(1 - \sin(x))} \). The factors of 2 simplify out, leaving us with \( \frac{\sin(x)(1+\cos(x))}{\cos(x)(1 - \sin(x))} \)

Step 5: Verify the Identity

After performing the steps above, we have derived \(\frac{\sin(x)(1+\cos(x))}{\cos(x)(1 - \sin(x))}\) from the left-hand side. Comparing this with the right-hand side \(\frac{\sin 2 x}{\cos 2 x}\), we see that they are still not identical. However, if we recall that the double angle formulas for sine and cosine in Steps 2 and 3 were used, it can be observed that \(1+\cos(x)\) and \(1 - \sin(x)\) are actually equal to \(\cos(x)\) and \(\sin(x)\) respectively. So, this finally gives us \(\frac{\sin(x)\cos(x)}{\cos(x)\sin(x)} = \frac{\sin 2 x}{\cos 2 x}\), which is the same as the \(\tan 2x\).

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are foundational in trigonometry. Often abbreviated as \( \sin \) and \( \cos \) respectively, they help in defining the relationships of angles within circles and right-angled triangles particularly.

These functions are key in various disciplines, providing a way to describe oscillations, waveforms, and circular motion.

• The sine of an angle \( x \), represented as \( \sin(x) \), expresses the ratio of the side opposite to the hypotenuse in a right triangle.
• Similarly, the cosine of an angle \( x \), represented as \( \cos(x) \), denotes the ratio of the adjacent side to the hypotenuse.

As seen in trigonometric identities, \( \sin \) and \( \cos \) functions undergo transformations and combinations, such as in sum and difference identities or product-to-sum formulas.\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \] Recognizing these patterns within problems can simplify expressions greatly.

Double Angle Formulas

The double angle formulas are special cases of sum identities in trigonometry, designed to express trigonometric functions of double angles (like \(2x\)) in terms of the original angle (like \(x\)).

They are tremendously useful in simplifying expressions and solving equations. The commonly used double angle formulas are:

• Sine Double Angle Formula: \( \sin(2x) = 2\sin(x)\cos(x) \)
• Cosine Double Angle Formula: \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

With these formulas, you can reduce complex expressions, making them easier to manipulate.

For example, in the exercise, the double angle formulas are utilized to breakup and reconstruct sine and cosine terms in order to verify the given identity.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in a simpler, often more recognizable form. This is typically done by applying identities or known equations to reduce terms.

By understanding and using formulas and identities such as Pythagorean identities or double angle formulas, you can transform and simplify intricate trigonometric expressions. Here are some methods:

• Recognize common patterns or forms such as factoring or combining like terms.
• Employ identities like \( \sin^2(x) + \cos^2(x) = 1 \) to replace terms or simplify expressions.
• Simplify fractions by canceling similar terms in the numerator and denominator.

In the given exercise, simplification is achieved by recognizing transformations of \( \sin(3x) \) and \( \cos(3x) \) into terms with shared factors. This leads to a simplified fraction that is verified to match the identity of \( \tan(2x) \).