Problem 60 - Precalculus Mathematics for Calculus

Question

(a) Express the function in terms of sine only. (b) Graph the function. $$f(x)=\sin x+\cos x$$

Step-by-Step Solution

Answer

(a) \( f(x) = \sqrt{2} \sin(x + \frac{\pi}{4}) \). (b) Graph a sine wave with an amplitude of \( \sqrt{2} \) and shifted \(-\frac{\pi}{4}\).

Step 1: Express Cosine in Terms of Sine

Use the identity \( \cos x = \sqrt{1 - \sin^2 x} \). Substitute this into the given function to express it in terms of sine: \[ f(x) = \sin x + \sqrt{1 - \sin^2 x} \].

Step 2: Derive a Sine-Only Function

To make the function a complete expression in terms of sine, let's rewrite the identity inside a trigonometric transformation. Note that the function can be rewritten as: \[ f(x) = \sqrt{2} \left( \frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} \right) \]. This is equivalent to \( \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \). Thus, you can express the function as: \[ f(x) = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \].

Step 3: Simplify and Verify the Identity

Verify that \( \sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \) by using the sum of angles formula. The sine of the sum is given by: \[ \sin \left( x + \frac{\pi}{4} \right) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} \].Simplifying, you know \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), so: \[ \sin \left( x + \frac{\pi}{4} \right) = \frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} \]. Multiply by \( \sqrt{2} \) to obtain: \[ \sqrt{2} \left( \frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} \right) = \sin x + \cos x \], confirming the expression.

Step 4: Graph the Function

Use the sine-only expression \( f(x) = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \) to graph the function. This is a standard sine curve with amplitude \( \sqrt{2} \), a horizontal shift of \(-\frac{\pi}{4}\), and period \( 2\pi \). Draw the sine curve accordingly, starting with the shift, setting peaks at \( \sqrt{2} \), and troughs at \(-\sqrt{2} \) on the y-axis.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used in numerous mathematical concepts. Its basic form is expressed as \( \sin x \). This function represents the y-coordinate of a point on the unit circle, where \( x \) is the angle in radians, measured from the positive x-axis. Sn highlights include:

• The sine function is periodic with a period of \( 2\pi \), which means it repeats its wave-like pattern every \( 2\pi \) units along the x-axis.
• It has an amplitude of 1, meaning the maximum and minimum values the function can reach are 1 and -1, respectively.
• It crosses the origin point, with key points
  • at \( \sin(0) = 0 \),
  • \( \sin(\pi/2) = 1 \),
  • \( \sin(\pi) = 0 \),
  • and \( \sin(3\pi/2) = -1 \).

Understanding the sine function is crucial in transforming and graphing more complex trigonometric functions.

Cosine Function

The cosine function, denoted by \( \cos x \), is another essential trigonometric function, paired closely with the sine function. It represents the x-coordinate of a point on the unit circle. This function has unique features that make it indispensable:

• Like the sine function, cosine is also periodic, with a period of \( 2\pi \).
• It shares the same range as the sine function, ranging from -1 to 1.
• Cosine starts at \( \cos(0) = 1 \), and key points along the unit circle include
  • \( \cos(\pi/2) = 0 \),
  • \( \cos(\pi) = -1 \),
  • and \( \cos(3\pi/2) = 0 \).
• The cosine graph is similar to the sine graph but shifted horizontally by \( \pi/2 \).

Combining the sine and cosine functions allows us to apply various transformations, crucial for expressing trigonometric functions in alternative forms.

Sum of Angles Formula

The sum of angles formula is a powerful tool in trigonometry that helps express the sine and cosine of angle sums. For instance, you can write the sine of the sum of two angles \( a \) and \( b \) as:\[\sin(a + b) = \sin a \cos b + \cos a \sin b.\]This formula helps transform and simplify trigonometric expressions. It plays a key role in the problem we've explored, allowing us to verify that \( \sin x + \cos x \) can be re-written using the compound angle formula. The step-by-step solution used the formula:

• The equivalent expression \( \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \) emerges from applying the sum of angles formula.
• This new expression simplifies the function by representing it solely in terms of the sine function with a phase shift (horizontal translation).

Recognizing the value of angle addition formulas simplifies complex trigonometric identities and equations.

Sine Graph

A sine graph is a visual representation of the sine function. It's a very instructive tool to help visualize and understand periodic properties, amplitude, and phase shifts. Here are some aspects to focus on:

• In its simplest form, a sine graph oscillates above and below the x-axis.
• The normal amplitude is 1, but transformations such as those in our exercise indicated an amplitude change to \( \sqrt{2} \).
• The graph we've analyzed has a phase shift determined by the addition of \( \frac{\pi}{4} \) inside the sine function. This results in the graph shifting horizontally to the left by \( \frac{\pi}{4} \).
• The period of the standard sine graph remains \( 2\pi \).
• The sine graph's regular peaks and troughs help predict values at given intervals, useful in applications ranging from physics to engineering.

By modifying the basic sine graph, you gain an intuitive understanding of transformations, laying a foundation for tackling complex trigonometric functions.