Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \sin \left(\theta+\frac{\pi}{4}\right)=\frac{1}{2} $$

Step-by-Step Solution

Answer

Solutions are \(\theta = -\frac{\pi}{12} + 2k\pi\) and \(\theta = \frac{7\pi}{12} + 2k\pi\), where \(k\) is an integer.

Step 1: Use the General Solution for Sine

To solve \(\sin(x) = \frac{1}{2}\), we need to identify the general angles where this equality holds. The sine function equals \(\frac{1}{2}\) at \(x = \frac{\pi}{6} + 2k\pi\) and \(x = \frac{5\pi}{6} + 2k\pi\), where \(k\) is any integer.

Step 2: Substitute and Solve for Theta

Substitute \(x = \theta + \frac{\pi}{4}\) into the equations from Step 1. This gives us two equations: \(\theta + \frac{\pi}{4} = \frac{\pi}{6} + 2k\pi\) and \(\theta + \frac{\pi}{4} = \frac{5\pi}{6} + 2k\pi\).

Step 3: Solve the First Equation

Subtract \(\frac{\pi}{4}\) from both sides to isolate \(\theta\) in the first equation: \(\theta = \frac{\pi}{6} - \frac{\pi}{4} + 2k\pi = -\frac{\pi}{12} + 2k\pi\).

Step 4: Solve the Second Equation

Subtract \(\frac{\pi}{4}\) from both sides to isolate \(\theta\) in the second equation: \(\theta = \frac{5\pi}{6} - \frac{\pi}{4} + 2k\pi = \frac{7\pi}{12} + 2k\pi\).

Step 5: Write the General Solution

Combine the results from Steps 3 and 4 to express the solutions of the original equation: \(\theta = -\frac{\pi}{12} + 2k\pi\) or \(\theta = \frac{7\pi}{12} + 2k\pi\), where \(k\) is an integer.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often denoted as \( \sin(x) \). It is particularly important in the study of periodic phenomena, wave motion, and circular motion. The function takes an angle as input and outputs the y-coordinate of the corresponding point on the unit circle.

Key properties of the sine function include:

• It is periodic with a period of \( 2\pi \), meaning that \( \sin(x) = \sin(x + 2\pi) \) for any angle \( x \).
• The range of the sine function is \([-1, 1]\), as it represents the projection of a point on the unit circle onto the y-axis.
• Common angles and their sine values include \( \sin(0) = 0 \), \( \sin(\pi/2) = 1 \), \( \sin(\pi) = 0 \), and \( \sin(3\pi/2) = -1 \).

Understanding these properties helps in solving problems involving the sine function, like finding angles that satisfy given trigonometric equations.

General Solution

The general solution for a trigonometric equation involves finding all possible angles that satisfy the given equation. Trigonometric functions are periodic, so multiple angles, apart from a basic angle, can satisfy the same equation.

To determine the general solution for equations like \( \sin(x) = \frac{1}{2} \), we first find a reference angle where this is true within one cycle (between 0 and \( 2\pi \)). For \( \sin(x) = \frac{1}{2} \), the reference angles are \( x = \pi/6 \) and \( x = 5\pi/6 \).

Since sine is periodic with a period of \( 2\pi \), the general solution extends these angles to all their equivalents by adding integer multiples of the period, \( 2k\pi \), where \( k \) is an integer. Therefore, the general solution is:

• \( x = \frac{\pi}{6} + 2k\pi \)
• \( x = \frac{5\pi}{6} + 2k\pi \)

This approach reveals all possible angles that satisfy the trigonometric equation.

Angle Measurement

Angle measurement is pivotal in trigonometry and comes in two primary units: degrees and radians. In mathematics, radians are the preferred unit because they simplify many trigonometric calculations.

Radians are the measure of the angle corresponding to the arc length of a circle's radius on its circumference. One complete revolution around a circle equals \( 2\pi \) radians. Hence, an angle of \( 180^\circ \) is equivalent to \( \pi \) radians, and \( 90^\circ \) is equivalent to \( \pi/2 \) radians.

Conversion between degrees and radians involves the relationships:

• \( 180^\circ = \pi \) radians
• \( 1^\circ = \pi/180 \) radians
• \( 1 \text{ radian} = 180/\pi \) degrees

Understanding and converting between these units is essential, particularly in contexts where the angle inputs and outputs need to be aligned with specific unit requirements.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals, known as periods. In trigonometry, functions like sine, cosine, and tangent are classic examples of periodic functions.

The sine function is periodic with a period of \( 2\pi \), which means it repeats every \( 2\pi \) units along the x-axis. This property is integral for creating general solutions to trigonometric equations.

Other key features of periodic functions include:

• Repetition of values over distinct intervals
• The shape of the graph remains unchanged as it moves along its period
• Applications are vast, from modeling natural phenomena like tides to engineering signals in telecommunications

Visualizing the repetitive pattern of periodic functions on a graph helps in understanding how solutions extend beyond the initial cycle, providing a comprehensive view of all possibilities.