Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 6 \cos ^{2} \theta+5 \cos \theta-4=0 $$

Step-by-Step Solution

Answer

The exact solution set is \( \theta = 60^\circ, 300^\circ \).

Step 1: Identify the Quadratic Form

The given equation is \( 6\cos^2 \theta + 5\cos \theta - 4 = 0 \). This equation is quadratic in form with respect to \( \cos \theta \).

Step 2: Substitute Variable for Simplicity

Let \( x = \cos \theta \). The equation becomes \( 6x^2 + 5x - 4 = 0 \).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 6 \), \( b = 5 \), and \( c = -4 \).

Step 4: Calculate the Discriminant

Compute the discriminant: \( b^2 - 4ac = 5^2 - 4 \times 6 \times (-4) = 25 + 96 = 121 \).

Step 5: Solve for \( x

The solutions for \( x \) are given by \( x = \frac{-5 \pm \sqrt{121}}{12} \). Evaluating gives: \( x_1 = \frac{-5 + 11}{12} = \frac{6}{12} = \frac{1}{2} \) and \( x_2 = \frac{-5 - 11}{12} = \frac{-16}{12} = \frac{-4}{3} \).

Step 6: Verify Solutions Within Domain

Since \( \cos \theta \) must be between -1 and 1, only \( x_1 = \frac{1}{2} \) is valid. \( x_2 = \frac{-4}{3} \) is outside the valid range for \( \cos \theta \).

Step 7: Find Angles for Valid Solution

For \( \cos \theta = \frac{1}{2} \), the angles are \( \theta = 60^\circ \) and \( \theta = 300^\circ \) (since \( \cos \theta \) is positive in the first and fourth quadrants).

Step 8: Solution Set

The solution set is \( \theta = 60^\circ, 300^\circ \). These are the values of \( \theta \) within \( 0^\circ \leq \theta < 360^\circ \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is a function that relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. When you see \( \cos \theta \), it refers to the cosine of the angle \( \theta \).

• In unit circle terms, \( \cos \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
• The cosine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.

The range of the cosine function is between -1 and 1. For instance, when solving equations like \( 6 \cos^2 \theta + 5 \cos \theta - 4 = 0 \), it's crucial that the solutions for \( \cos \theta \) remain within this range. These limitations help when validating solutions, as seen in this exercise where \( x_2 = \frac{-4}{3} \) was discarded because it does not fall within -1 to 1.

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation has the form \( ax^2 + bx + c = 0 \).

• The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) finds the values of \( x \) that make the equation zero.
• The part under the square root, \( b^2 - 4ac \), is called the discriminant and determines the number and type of roots.

The quadratic formula applies to any quadratic equation and provides exact solutions even when it's hard to factor the expression. In our context, when dealing with trigonometric functions like cosine, solving \( 6 \cos^2 \theta + 5 \cos \theta - 4 = 0 \) involves treating \( \cos \theta \) as an algebraic variable. By substituting \( x = \cos \theta \), we converted the equation to a standard quadratic form \( 6x^2 + 5x - 4 = 0 \) so that we could apply the formula effectively.

Trigonometric Identities

Trigonometric identities are essential tools that help simplify trigonometric expressions and equations. Some of the most commonly used include:

• Basic identities like \( \cos^2\theta + \sin^2\theta = 1 \) can help transform and solve equations.
• They can show relationships between different trigonometric functions, such as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).

These identities not only simplify calculations but also clarify the relationships between different angles and sides in triangles. In our exercise, understanding the range of the cosine function \( -1 \leq \cos \theta \leq 1 \) as part of the fundamental trigonometric identity helps evaluate the validity of solutions and find accurate angle measures for \( \theta \). Overall, knowledge of trigonometric identities is essential to scrutinize every step and ensure the solution set lies within a valid trigonometric context.