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Magazine Chapter 6: Problem 46
\(\cos ^{2} 3 \theta-6 \cos 3 \theta+4=0\)
Short Answer
Expert verified
The valid solution is \( 3\theta = \cos^{-1}(3 - \sqrt{5}) \).
Step by step solution
01
Recognize the quadratic form
The equation \( \cos^2 3\theta - 6 \cos 3\theta + 4 = 0 \) is in the form of a quadratic: \( ax^2 + bx + c = 0 \), where we let \( x = \cos 3\theta \). This simplifies our expression to \( x^2 - 6x + 4 = 0 \).
02
Calculate the discriminant
Find the discriminant \( D \) of the quadratic equation \( x^2 - 6x + 4 = 0 \) using \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = -6 \), and \( c = 4 \), so \( D = (-6)^2 - 4 \cdot 1 \cdot 4 = 36 - 16 = 20 \).
03
Apply the quadratic formula
Since the discriminant is positive, we use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \). Substitute \( a = 1 \), \( b = -6 \), and \( D = 20 \), this gives \( x = \frac{6 \pm \sqrt{20}}{2} \). Simplify further to \( x = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5} \).
04
Verify valid solutions for cosine
The cosine function has a range \([-1, 1]\). Let's check if \( 3 + \sqrt{5} \) and \( 3 - \sqrt{5} \) lie within this range.\(-1 \leq \cos 3\theta \leq 1\). Evaluate the expression: \( 3 - \sqrt{5} \) is approximately \( 0.764 \), which is a valid cosine value; but \( 3 + \sqrt{5} \approx 5.236 \), which is not valid.
05
Solve for \(3\theta\) when \(\cos 3\theta = 3 - \sqrt{5}\)
Use the fact that \( \cos 3\theta = 3 - \sqrt{5} \). Find the angle \( 3\theta \) using \( \cos^{-1}(3 - \sqrt{5}) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is an algebraic expression of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. This structure is pivotal in various mathematical problems, due to the many methods available for finding their roots.
To solve a quadratic equation, you can apply several techniques:
To solve a quadratic equation, you can apply several techniques:
- **Factoring:** This involves expressing the quadratic equation as a product of its linear factors.
- **Completing the Square:** This method transforms the equation into a perfect square trinomial.
- **Using the Quadratic Formula:** This is a direct approach where roots are found using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Cosine Function
The cosine function is one of the fundamental trigonometric functions, commonly denoted as \( \cos(\theta) \). It measures the adjacent side over the hypotenuse in a right triangle and applies to angle measurement in radians or degrees.
The cosine function has a few critical properties:
The cosine function has a few critical properties:
- **Periodic Nature:** The cosine function repeats every \( 2\pi \) radians or 360 degrees, which is its period.
- **Range:** The values of cosine lie between -1 and 1, inclusive. This range is critical when checking valid outputs for trigonometric equations.
- **Even Function:** This means \( \cos(-\theta) = \cos(\theta) \), implying symmetry about the y-axis.
Discriminant in Quadratic Equations
The discriminant of a quadratic equation, given by \( D = b^2 - 4ac \), is essential in understanding the nature of its roots. It provides insight without needing to solve the equation fully.
- **Positive Discriminant:** When \( D > 0 \), the quadratic equation has two distinct real roots. In our example, a positive discriminant (\( 20 \)) suggested possible real solutions for \( \cos 3\theta \).
- **Zero Discriminant:** If \( D = 0 \), the equation has one real root, also known as a repeated root.
- **Negative Discriminant:** Here, \( D < 0 \), indicating two complex roots, not real numbers.
