Problem 54 - Algebra and Trigonometry

Question

\(39-56 \approx\) Solve the given equation. $$ \tan \theta \sin \theta+\sin \theta=0 $$

Step-by-Step Solution

Answer

Solutions are \(\theta = n\pi\) and \(\theta = \frac{3\pi}{4} + n\pi\) for integer \(n\).

Step 1: Factor the Equation

The given equation is \(\tan \theta \sin \theta + \sin \theta = 0\). Notice that \(\sin \theta\) is a common factor in both terms. Factor out \(\sin \theta\) from the equation: \( \sin \theta ( \tan \theta + 1 ) = 0 \).

Step 2: Solve for Sine

When \(\sin \theta = 0\), the solutions for \(\theta\) occur at integer multiples of \(\pi\). Therefore, the solutions are \(\theta = n\pi\), where \(n\) is an integer.

Step 3: Solve for Tangent

For the factor \(\tan \theta + 1 = 0\), solve for \(\theta\). \(\tan \theta = -1\) implies \(\theta = \frac{3\pi}{4} + n\pi\), where \(n\) is an integer.

Step 4: Combine Solutions

Combine all solutions from the individual factors. The general solutions are \(\theta = n\pi\) and \(\theta = \frac{3\pi}{4} + n\pi\) for any integer \(n\).

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles in triangles. It is especially concerned with right-angled triangles, where it helps in calculating unknown sides and angles using known values. Some of the most important functions in trigonometry include the sine, cosine, and tangent functions, which are tools used to describe the ratios of the sides of a right-angled triangle relative to its angles. The core elements of trigonometry can be summarized as follows:

• The hypotenuse, which is the longest side in a right-angled triangle.
• The opposite side, which is opposite the angle of interest in a right triangle.
• The adjacent side, which is next to the angle of interest in a right triangle.

Trigonometry is not only about solving triangles. It also provides insights and tools for understanding waves, oscillations, and many real-world applications in fields like physics and engineering.

Diving into the Sine Function

The sine function is one of the primary trigonometric functions, often denoted as \sin(\theta)\. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Mathematically, it is expressed as:\[ \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]This function is periodic, meaning it repeats its values in regular intervals. Its period is \(2\pi\), which means every \(2\pi\) units along the x-axis, the sine function starts its cycle again. Some notable properties of the sine function include:

• Its range is \([-1, 1]\).
• At \(\theta = 0, \pi\), and \(2\pi\), \sin(\theta)\ equals 0.
• It reaches its maximum value of 1 at \(\theta = \frac{\pi}{2}\) and \(\frac{5\pi}{2}\), and its minimum value of -1 at \(\theta = \frac{3\pi}{2}\).

The sine function plays a crucial role in solving trigonometric equations, as seen in exercises where it acts as a common factor to simplify expressions.

Exploring the Tangent Function

The tangent function is another fundamental trigonometric function, commonly represented as \tan(\theta)\. It is defined as the ratio of the sine to the cosine of an angle.The mathematical representation of the tangent function is:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]Tangent has a period of \(\pi\), which means it repeats every \(\pi\) units. Unlike sine and cosine, tangent can take any real number value, as its range is \(-\infty, \infty\).Here are a few important characteristics of the tangent function:

• It is undefined at angles where the cosine is zero (e.g., \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\)).
• It crosses zero at integer multiples of \(\pi\) (e.g., \(\theta = 0, \pi, 2\pi\)).
• The function has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\), causing the function to "jump" from positive to negative infinity.

Understanding the tangent function is crucial for solving trigonometric equations involving it, such as in scenarios where one needs to solve for \(\theta\) when \tan(\theta) + 1 = 0\, leading to solutions at specific angles.