Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find, to the nearest degree, all values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ \frac{\sin \theta}{2}=\frac{3}{\sin \theta+2} $$

Step-by-Step Solution

Answer

No real values of \(\theta\) satisfy the equation within the given interval.

Step 1: Clear the Denominators

First, let's eliminate the fractions by multiplying both sides by the common denominator, which is \(2(\sin \theta + 2)\): \[2(\sin \theta + 2)\times \frac{\sin \theta}{2} = 2(\sin \theta + 2)\times \frac{3}{\sin \theta + 2}\]This simplifies to: \[\sin \theta (\sin \theta + 2) = 6\]

Step 2: Expand and Rearrange

Expand the left side and rearrange all terms to one side of the equation:\[\sin^2 \theta + 2\sin \theta - 6 = 0\]

Step 3: Use the Quadratic Formula

The equation \(\sin^2 \theta + 2\sin \theta - 6 = 0\) is a quadratic equation in the form \(a x^2 + bx + c = 0\), where \(a = 1\), \(b = 2\), and \(c = -6\). Use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to solve for \(\sin \theta\):\[\sin \theta = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}\]Calculating the discriminant and simplifying:\[\sin \theta = \frac{-2 \pm \sqrt{4 + 24}}{2} = \frac{-2 \pm \sqrt{28}}{2}\]Further simplify \(\sqrt{28} = 2\sqrt{7}\):\[\sin \theta = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7}\]

Step 4: Determine Valid Solutions for Sin(θ)

Now, we have two potential solutions for \(\sin \theta\):1. \(\sin \theta = -1 + \sqrt{7}\)2. \(\sin \theta = -1 - \sqrt{7}\)Check if they are valid within the range \(-1 \leq \sin \theta \leq 1\): - \(\sin \theta = -1 + \sqrt{7} \approx 1.645 \), which is out of the valid range.- \(\sin \theta = -1 - \sqrt{7} \approx -3.645 \), which is also out of the valid range.Since neither solution is valid, we conclude that no real solutions exist for \(\theta\).

Key Concepts

Understanding Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine and cosine, and they are equations to be solved for angles. Understanding these equations is crucial for solving various problems in mathematics.

To solve a trigonometric equation, such as the one given in the exercise, we first need to isolate the trigonometric part by eliminating fractions or other terms. This often involves clearing denominators through multiplication.

• Multiplying both sides by the least common denominator (LCD) clears fractions.
• Rearranging terms helps in expressing the equation in a standard form.

For example, consider the given equation: \(\frac{\sin \theta}{2} = \frac{3}{\sin \theta+2}.\)Clearing the denominator by multiplying through by \(2(\sin \theta + 2)\) allows us to transform the problem into a more manageable polynomial form, where we can then apply quadratic solution techniques.

Exploring Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable involved. They are critical tools for transforming and simplifying trigonometric equations.

Some common trigonometric identities used frequently include:

• Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
• Angle Sum and Difference Identities
• Double Angle and Half Angle Identities

These identities can play a significant role in simplifying complex equations to forms where other mathematical strategies, like the quadratic formula, become applicable. In the exercise, once the trigonometric equation was transformed into a quadratic equation \(\sin^2 \theta + 2\sin \theta - 6 = 0,\)understanding these identities was essential for further simplification.

The Role of the Discriminant in Quadratic Equations

The discriminant is a vital part of the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

It determines the nature of the roots of a quadratic equation. The discriminant is given by \(b^2 - 4ac\).

• A positive discriminant indicates two distinct real solutions.
• A zero discriminant indicates one real solution.
• A negative discriminant indicates no real solutions, only complex ones.

In the provided exercise, calculating the discriminant was crucial:\(2^2 - 4 \cdot 1 \cdot (-6) = 28.\)The positive value led us to solutions that were checked against the validity for \(\sin \theta\),but neither was within the range \([-1, 1]\).This illustrates that while solutions for \(x\) may exist, they may not all be physically viable in trigonometric terms, emphasizing the importance of the discriminant in determining solution viability.