Problem 59 - Algebra and Trigonometry

Question

(a) Express the function in terms of sine only. (b) Graph the function. $$ g(x)=\cos 2 x+\sqrt{3} \sin 2 x $$

Step-by-Step Solution

Answer

(a) \( g(x) = 2\sin(2x + \frac{\pi}{3}) \). (b) Graph has amplitude 2, period \( \pi \), phase shift \( -\frac{\pi}{3} \).

Step 1: Use Double Angle Identities

The given function is \( g(x) = \cos 2x + \sqrt{3} \sin 2x \). To express the function in terms of sine only, we'll use double angle identities. Recall that \( \cos 2x \) can be expressed as \( \cos^2 x - \sin^2 x \) or \( 1 - 2\sin^2 x \) or \( 2\cos^2 x - 1 \). However, there's not a direct analogous identity for \( \cos 2x \) to express in terms of sine only unless mixed with sine.

Step 2: Write Cosine as a Linear Combination of Sine

To express \( \cos 2x + \sqrt{3} \sin 2x \) only in terms of sine, you can write it in the form of \( R \sin(2x + \phi) \). This technique requires converting \( a \cos 2x + b \sin 2x \) into \( R \sin(2x + \phi) \) where \( R = \sqrt{a^2 + b^2} \) and \( \tan \phi = \frac{b}{a} \). In our case, \( a = 1 \) and \( b = \sqrt{3} \).

Step 3: Determine the Amplitude and Phase Shift

Calculate \( R \): \[ R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \]Determine \( \phi \) such that \( \tan \phi = \frac{\sqrt{3}}{1} = \sqrt{3} \). Since \( \tan \phi = \sqrt{3} \), \( \phi \) corresponds to \( 60^\circ \) or \( \frac{\pi}{3} \) radians. Thus, \( g(x) = 2 \sin(2x + \frac{\pi}{3}) \).

Step 4: Graph the Function

To graph the function \( g(x) = 2\sin(2x + \frac{\pi}{3}) \), start by identifying the key components:- The amplitude is 2.- The period is \( \frac{2\pi}{2} = \pi \).- The phase shift is \( -\frac{\pi}{3} \), as the function shifts left by \( \frac{\pi}{3} \).Draw the sine wave starting from \( -\frac{\pi}{3} \) and complete a cycle every \( \pi \), ranging from -2 to 2.

Key Concepts

Double Angle Identities

Double angle identities are a fundamental component of trigonometry that allow us to express trigonometric functions at twice the angle of a known function. The double angle identity for cosine is particularly versatile and can be expressed in several forms, including:

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

These identities are handy because they enable the transformation of trigonometric expressions into more workable forms.
In the given problem, to express the cosine and sine functions in terms of sine only, involves writing the expression as a linear combination: \( R \sin(2x + \phi) \).
This solution simplifies the graphing of these expressions by presenting a standard sine wave, defined by its amplitude and phase shift.

Sine Function

The sine function is one of the core trigonometric functions, defined as the ratio of the opposite side to the hypotenuse in a right triangle. In the context of graphing, it exhibits a wave-like shape known as a sine wave.

• The standard sine function is expressed as \( \sin x \) and oscillates between -1 and 1.
• The amplitude of a sine wave is the distance from the middle of the wave to its peak. In \( g(x) = 2 \sin(2x + \frac{\pi}{3}) \), the amplitude is 2, meaning the wave reaches from -2 to 2.
• The function's frequency and period are adjusted based on the coefficient of \( x \) inside the function. Here, \( 2x \) indicates the function completes its cycle twice as fast as a regular sine wave with a period \( \pi \).

Thus, understanding the sine function helps in interpreting and transforming other trigonometric expressions for different applications.

Trigonometric Graphing

When graphing trigonometric functions like \( g(x) = 2\sin(2x + \frac{\pi}{3}) \), several key characteristics must be considered:

• Amplitude: The amplitude of this function is 2, stretching the sine wave vertically.
• Period: The period of the function is calculated as \( \frac{2\pi}{2} = \pi \), indicating the horizontal length for one full wave cycle.
• Phase Shift: Represented by \( \frac{\pi}{3} \), the phase shift moves the entire wave to the left by \( \frac{\pi}{3} \) units.

Graphing begins by plotting these values on a coordinate plane, marking the critical points where the wave reaches its maximum, minimum, and zero crossings.
This graphing method effectively reveals how modified trigonometric functions behave and aids in visualizing shifts and transformations of sine waves in response to changes in function parameters.