Problem 55 - Algebra and Trigonometry

Question

Write the expression in terms of sine only. \(-\sqrt{3} \sin x+\cos x\)

Step-by-Step Solution

Answer

The expression is rewritten as \(2 \cdot sin\left(x - \frac{\pi}{6}\right)\) in terms of sine only.

Step 1: Identify the Trigonometric Identity

We know that the basic trigonometric identity relating sine and cosine is:\[sin^2 x + cos^2 x = 1\]However, to express cosine in terms of sine, we use:\[cos x = \sqrt{1 - sin^2 x}\]In this problem, we need another identity for expressing both functions in terms of sine.

Step 2: Express Cosine in Terms of Sine Using a Phase Shift

We recognize that the expression can be rewritten by factoring with common magnitudes. Notably, we can introduce a trigonometric identity to match multiples of sine and cosine:\[a \cdot sin x + b \cdot cos x = R \cdot sin(x + \phi)\]where \( R = \sqrt{a^2 + b^2} \). First, let's calculate the magnitude \( R \).

Step 3: Calculate the Magnitude R

For the expression \(-\sqrt{3} sin x + cos x\), set \( a = -\sqrt{3} \) and \( b = 1 \). Thus, the magnitude \( R \) is:\[R = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2\]Now, we have:\[-\sqrt{3} sin x + cos x = 2 \cdot sin(x + \phi)\]

Step 4: Determine the Phase Shift φ

The phase shift \( \phi \) is determined by:\[cos \phi = \frac{a}{R} = \frac{-\sqrt{3}}{2} \quad \text{and} \quad sin \phi = \frac{b}{R} = \frac{1}{2}\]The angle \( \phi \) that satisfies these conditions is \( x - \frac{\pi}{6} \); hence:\[-\sqrt{3} sin x + cos x = 2 \cdot sin\left(x - \frac{\pi}{6}\right)\]

Step 5: Express in Terms of Sine Only

Now, since we have calculated that:\[-\sqrt{3} sin x + cos x = 2 \cdot sin\left(x - \frac{\pi}{6}\right)\]Thus, the entire expression is rewritten entirely in terms of \( sin \).

Key Concepts

Trigonometric Identities

When dealing with trigonometric problems, it's crucial to have a firm grasp on trigonometric identities. They are mathematical equations that relate the trigonometric functions to each other.
These identities enable us to transform and simplify expressions for easier calculation or insight. The most fundamental among these is the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
This relationship forms a basis to explore further identities.Some other key identities are:

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

These identities are powerful tools that allow for the transformation of expressions, particularly important when we wish to express one trigonometric function in terms of another, like how cosine can be written in terms of sine through the equation \( \cos x = \sqrt{1 - \sin^2 x} \). Understanding these identities can greatly simplify complex problems.

Phase Shift

Phase shift is a method to modify a trigonometric function by shifting it horizontally along the x-axis.
This plays an essential role in expressing a combination of sine and cosine functions in terms of just one function, usually sine or cosine.
For instance, in the expression \( a \sin x + b \cos x \), a phase shift can be used to simplify it to \( R \sin(x + \phi) \).In this process, the expression is transformed where:

• \( R = \sqrt{a^2 + b^2} \) is the magnitude, providing the amplitude of the new single trigonometric function.
• \( \phi \), the phase angle, shifts the function horizontally.

To find \( \phi \), we use the relations \( \cos \phi = \frac{a}{R} \) and \( \sin \phi = \frac{b}{R} \). It gives us the specific angle by which the sine wave is shifted. Understanding and applying phase shift effectively helps to merge multiple trigonometric functions into a simpler form.

Express in Terms of Sine

Expressing an expression in terms of sine alone involves rewriting any combination of sine and cosine into a single sine function. This process simplifies calculations and interpretation, especially if the task requires only sine.
For the given expression \(-\sqrt{3} \sin x + \cos x\), we aimed to transform it to a form like \( R \sin(x + \phi) \).Here’s how it's done:

• Calculate \( R \) as the resultant magnitude: \( R = \sqrt{a^2 + b^2} \).
• Determine the phase shift \( \phi \) using \( \cos \phi \) and \( \sin \phi \) relations.

Once these are known, the original expression can be rewritten. For this example, it becomes \( 2 \sin(x - \frac{\pi}{6}) \), thereby fully expressing it as a sine function.
Mastering this technique is crucial for making complex trigonometric expressions manageable, and it enhances understanding of the underlying patterns in trigonometric equations.