91Ó°ÊÓ

Let's begin with the quadratic formula. This essential tool helps us solve quadratic equations, which have the form: \[ ax^2 + bx + c = 0 \] The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are coefficients from the equation. By plugging these values into the formula, we can find the roots or solutions of the quadratic equation. Remember, the expression inside the square root, \(b^2 - 4ac\), is called the discriminant and determines the nature of the roots. If - The discriminant is positive, there are two real and distinct solutions. - It is zero, there is one real and repeated solution. - It is negative, there are two complex solutions. In our exercise, the equation is expressed in terms of \( \cos \phi\), making it a quadratic equation in disguise. By identifying \(a = 6\), \(b = 5\), and \(c = 1\), we can apply the quadratic formula to solve for \( \cos \phi\). This substitution is a common and useful step when dealing with trigonometric equations that resemble quadratic form.

The cosine function, \( \cos \), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. It's defined as: \[ \cos \theta = \frac{adjacent}{hypotenuse} \] For any angle \( \theta \), \( \cos \theta \) produces a value between -1 and 1. This function is periodic with a period of \(2\pi\) radians, meaning it repeats every \(2\pi\) units. The related graph, called a cosine wave, fluctuates above and below the x-axis. Key properties of the cosine function include: - \( \cos(0) = 1 \) - \( \cos(\pi / 2) = 0 \) - \( \cos(\pi) = -1 \) - \( \cos(3\pi / 2) = 0 \) In trigonometry, the cosine function helps us solve for unknown angles when the cosine value is known. In our example, after solving the quadratic equation, we found two possible values for \( \cos \phi \): -\(1/3\) and -\(1/2\). Using inverse cosine function, or \( \cos^{-1} \), allows us to determine the specific angles \( \phi \) that correspond to these cosine values.

Once we have the values for \( \cos \phi \), the next step is to identify the corresponding angles within a specified interval. Here, we need to find all solutions in the interval \( [0, 2\pi) \) or \( [0^{\circ}, 360^{\circ}) \). To do this, we use the inverse cosine function: - For \( \cos \phi = -1/3 \), find \( \phi = \cos^{-1}(-1/3) \), which yields approximate solutions \( \phi \approx 1.9106 \) and \( \phi \approx 4.3726 \). - For \( \cos \phi = -1/2 \), find \( \phi = \cos^{-1}(-1/2) \), which yields solutions \( \phi = 2\pi/3 \approx 2.0944 \) and \( \phi = 4\pi/3 \approx 4.1888 \). These angles represent the critical points where the cosine values we calculated occur. By carefully examining our calculator or unit circle, we can determine that each value corresponds to two angles within one complete cycle (\(0\) to \(2\pi\)). Verifying these solutions graphically, as suggested in the original exercise, confirms their accuracy and ensures thorough understanding and correctness.