Problem 65 - Algebra 2

Question

Find the complete solution of \(\sin ^{2} \theta+2 \sin \theta+1=0 .\) (Hint: How would you solve \(x^{2}+2 x+1=0 ? )\)

Step-by-Step Solution

Answer

The solution is \(\theta=3\pi/2 + 2\pi n\), where \(n\) is any integer.

Step 1: Simplify the quadratic equation

The quadratic equation \(x^{2}+2 x+1=0\) can be written as \((x+1)^{2}=0\). Setting this equal to zero, we find that \(x+1=0\) meaning \(x=-1\).

Step 2: Apply quadratic solution to trigonometric equation

Similarly, the equation \(\sin ^{2} \theta+2 \sin \theta+1=0\) can be written as \((\sin \theta+1)^{2}=0\). Applying the same process, we get \(\sin \theta+1=0\), and therefore \(\sin \theta=-1\).

Step 3: Find angles using the sine function

Now we need to find what angles will give us a sine of -1. The angles that satisfy this condition are \(\theta=3\pi/2 + 2\pi n\), where \(n\) is any integer. This is the complete set of solutions.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, most commonly written in the form \(ax^2 + bx + c = 0\). Here's a quick breakdown of their components and solutions:

• The term \(ax^2\) is the quadratic term, and it usually dictates the shape of the parabola, which is the graph of a quadratic equation.
• Solving a quadratic equation like \(x^2 + 2x + 1 = 0\) involves finding the values of \(x\) that make the equation true. There are several methods to solve them, including factoring, using the quadratic formula, or completing the square.

For example, the equation \((x + 1)^2 = 0\) is a simplified, factored version of \(x^2 + 2x + 1 = 0\) and tells us that the solution is \(x = -1\). Solving such equations forms a strong foundation for tackling more complex mathematical problems.

Sine Function

The sine function is a basic trigonometric function that relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. Here's why it's useful:

• Sine functions are periodic, repeating every \(2\pi\) intervals in radian measure, or every 360 degrees.
• In solving trigonometric equations like \(\sin^2 \theta + 2\sin \theta + 1 = 0\), recognizing the familiar quadratic form allows us to apply techniques from algebra to trigonometry.
• Generally, finding the sine of an angle involves using known values from the unit circle, where the sine of \(\theta\) corresponds to the y-coordinate of the point intersecting the unit circle.

These principles guide us in finding solutions for trigonometric equations and serve as building blocks for further studies in mathematics.

Angle Solutions

Angle solutions in trigonometry often require finding specific angle measures that satisfy certain conditions, like those given by the sine, cosine, or tangent functions. In the context of this problem:

• We identify angles based on given trigonometric function values, such as \(\sin \theta = -1\) in this example.
• The angle \(\theta = \frac{3\pi}{2}\) is a standard angle where the sine value is \(-1\). The general form \(\theta = \frac{3\pi}{2} + 2\pi n\) covers all possible solutions, where \(n\) is any integer, representing the periodic nature of sine.
• In practice, this means that every full rotation (\(2\pi\)) from \(\frac{3\pi}{2}\) can be considered another solution, harnessing the cyclical behavior of trigonometric functions.

Understanding angle solutions is crucial for fully grasping how trigonometric functions behave over different intervals, an essential skill in both pure and applied mathematics.