Problem 11 - 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. $$ 3 \csc ^{2} \theta-2 \csc \theta=2 $$

Step-by-Step Solution

Answer

\( \theta \) is approximately \(118^\circ\) and \(242^\circ\).

Step 1: Set up the equation

First, we need to express the equation in terms of a single variable. Rewrite the given equation as follows: \( 3 ext{csc}^2 \theta - 2 ext{csc} \theta - 2 = 0 \). Here, \( \text{csc} \theta \) is the function of \( \theta \) that we need to solve for.

Step 2: Substitute variable for ease of solving

Let \( x = \text{csc} \theta \). Substituting \( x \) gives us the quadratic equation: \( 3x^2 - 2x - 2 = 0 \).

Step 3: Apply the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 3 \), \( b = -2 \), and \( c = -2 \) into the formula to get: \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} \].

Step 4: Simplify under the square root

Calculate the discriminant \( b^2 - 4ac \): \[ 4 - 24(-2) = 4 + 24 = 28 \].So the square root term simplifies to \( \sqrt{28} = 2\sqrt{7} \).

Step 5: Solve for variable x

Substitute the simplified discriminant back into the formula: \[ x = \frac{2 \pm 2\sqrt{7}}{6} \].Further simplify: \[ x = \frac{1 \pm \sqrt{7}}{3} \].

Step 6: Convert to angle values for \( \theta \)

Now, resolve for \( \theta \) using \( x = \text{csc} \theta \). Since \( \text{csc} \theta = \frac{1}{\sin \theta} \), we have \( \sin \theta = \frac{3}{1 + \sqrt{7}} \) and \( \sin \theta = \frac{3}{1 - \sqrt{7}} \), if valid. Check the ranges for these values to find approximate \( \theta \).

Step 7: Determine \( \theta \) in degrees

Using the valid sine inverses, find \( \theta \) within the given range \( 0^\circ \leq \theta < 360^\circ \). Since the values need to be in the range of the \( \sin \) function \((-1 \leq \sin \theta \leq 1)\), you compute to find only the valid \( \theta \): \( \theta \approx 118^\circ, 242^\circ \) for the first valid fraction.

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that relate the angles and sides of a triangle. These identities are very useful in simplifying expressions and solving equations involving trigonometric functions. In our exercise, we deal with the cosecant function, which is the reciprocal of the sine function:

• The cosecant identity is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).
• Other important identities include \( \sin^2 \theta + \cos^2 \theta = 1 \), which relates sine and cosine.
• The reciprocal identities, such as \( \sec \theta = \frac{1}{\cos \theta} \) and \( \cot \theta = \frac{1}{\tan \theta} \), help to rewrite equations in manageable forms.

In the given problem, we used the identity for cosecant to convert the problem of finding \( \theta \) into a more familiar quadratic equation. The goal was to isolate the trigonometric function to make use of the quadratic formula, which we'll discuss next.

Quadratic Formula

The quadratic formula provides a straightforward way to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula is very useful because it offers a solution with just a few straightforward substitutions: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the values \( a \), \( b \), and \( c \) come from the standard form of the quadratic equation. The solution may yield two real roots, one real root, or no real roots, depending on the discriminant \( b^2 - 4ac \). In our exercise, substituting in the appropriate numbers revealed:

• The discriminant calculation gives \( \sqrt{28} = 2\sqrt{7} \).
• Two potential solutions are possible, as \( x = \frac{1 \pm \sqrt{7}}{3} \).

This procedure then helps us uncover the values of \( x \) that correspond to \( \csc \theta \), setting the stage to convert these into angles using trigonometric functions.

Trigonometric Functions

Trigonometric functions describe the relationship between the angles and sides of right triangles. They include sine, cosine, tangent, cosecant, secant, and cotangent. Each has a specific definition:

• The sine function \( \sin \theta \) returns the ratio of the opposite side to the hypotenuse.
• The cosecant function \( \csc \theta = \frac{1}{\sin \theta} \) is used when determining \( \theta \) from its sine.
• The cosine and tangent functions relate other angle-side ratios, and their reciprocals \( \sec \theta \), \( \cot \theta \) offer alternative formulations.

In our exercise, once solutions for \( x = \csc \theta \) are found, they are inverted to find \( \sin \theta \). Validating \( \sin \theta \) is crucial since it must lie within \(-1 \leq \sin \theta \leq 1\). Calculating \( \theta \) involves inverses of these functions. The angle solutions in the range \( 0^\circ \leq \theta < 360^\circ \) emerged as \( 118^\circ \) and \( 242^\circ \), satisfying the trigonometric behavior constraints and displaying the cyclical nature of acoustic and trigonometric functions.