Problem 44 - Algebra and Trigonometry

Question

\(41-44=\) Write the expression in terms of sine only. $$ 3 \sin \pi x+3 \sqrt{3} \cos \pi x $$

Step-by-Step Solution

Answer

The expression is \(6 \sin(\pi x + \frac{\pi}{3})\) in terms of sine only.

Step 1: Analyze the Given Expression

We are given the expression \(3 \sin(\pi x) + 3 \sqrt{3} \cos(\pi x)\). Our task is to write this in terms of sine only.

Step 2: Use Sine Addition Formula

Using the sine addition formula, \(a \sin \theta + b \cos \theta = R \sin(\theta + \phi)\), where \(R = \sqrt{a^2 + b^2}\) and \(\tan \phi = \frac{b}{a}\), allows us to combine both sine and cosine into a single sine term.

Step 3: Calculate R

First, calculate \(R\) using \(a = 3\) and \(b = 3\sqrt{3}\):\[R = \sqrt{(3)^2 + (3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6\]

Step 4: Determine the Angle \(\phi\)

Next, find \(\tan \phi\) which is \(\frac{b}{a} = \frac{3\sqrt{3}}{3} = \sqrt{3}\). The angle \(\phi\) whose tangent is \(\sqrt{3}\) is \(\frac{\pi}{3}\) radians.

Step 5: Substitute Back into Sine Formula

Substitute \(R\) and \(\phi\) back into the sine addition formula:\[3 \sin(\pi x) + 3 \sqrt{3} \cos(\pi x) = 6 \sin(\pi x + \frac{\pi}{3})\]

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It describes how the sine angle relates to lengths in right-angled triangles.
The sine of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.
Mathematically, it is expressed as:

• \( ext{sin}( heta) = \frac{opposite}{hypotenuse} \)

This function oscillates between -1 and 1 as the angle varies.
The graph of the sine function is a smooth, wave-like periodic curve that repeats every \(2\pi\) radians. It's critical in understanding circular motion and waves.

Cosine Function

Similar to the sine function, the cosine function is also based on right-angled triangles. It compares the adjacent side's length to the hypotenuse.
The cosine of an angle is given by:

• \( ext{cos}( heta) = \frac{adjacent}{hypotenuse} \)

The cosine function also varies between -1 and 1 and has a periodic wave-like curve. It is just shifted in phase compared to the sine wave.
This shift makes cosine useful, especially when working with waveforms and oscillations in physics and engineering.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable.
These identities establish important relationships between functions like sine and cosine.
One key identity is the Pythagorean identity:

• \( ext{sin}^2( heta) + ext{cos}^2( heta) = 1 \)

Understanding such identities helps simplify and transform expressions.
They are instrumental in solving complex trigonometric equations and proving mathematical theorems.

Angle Transformation

Angle transformation is about changing the form of an expression involving angles from one function to another.
This process often uses angle sum formulas to represent functions like sine and cosine with transformed angles.A well-known transformation uses the sine addition formula:

• \( a \sin \theta + b \cos \theta = R \sin(\theta + \phi) \)

Here, \(R\) is calculated as \( \sqrt{a^2 + b^2} \) and \(\phi\) is determined using \(\tan \phi = \frac{b}{a}\).
Applying angle transformations allows us to convert and manipulate expressions efficiently, as shown in the original problem's solution.