Question

Solve each equation for all nonnegative values of \(x\) less than \(360^{\circ} .\) Do some by calculator. $$4 \cos ^{2} x+4 \cos x=-1$$

Short answer

The solution for \(x\) in the given range for the equation \(4 \cos^2 x + 4 \cos x = -1\) is \(x = 120^\circ\) and \(x = 240^\circ\).

Step-by-step solution

Simplify the Equation

Start by dividing the entire equation by 4 to simplify it: \[4 \cos^2 x + 4 \cos x = -1 \ \Rightarrow \cos^2 x + \cos x = -\frac{1}{4}\]

Apply Quadratic Formula

Let \( u = \cos x \). Then the equation becomes a quadratic equation: \[u^2 + u + \frac{1}{4} = 0 \] Now use the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = 1 \), and \( c = \frac{1}{4} \) to find the values of \(u\).

Calculate the Discriminant

Compute the discriminant \(\Delta = b^2 - 4ac\) which is \(1^2 - 4(1)(\frac{1}{4}) = 1 - 1 = 0\). The discriminant being 0 indicates that there is only one real solution.

Solve for u

Since the discriminant is 0, there is only one solution: \[u = \frac{-1}{2}\]This means \(\cos x = -\frac{1}{2}\).

Find the Values of x

Use a calculator, or the unit circle, to find the values of \(x\) where \(\cos x = -\frac{1}{2}\) with nonnegative values of \(x\) less than \(360^\circ\). The solutions are \(x = 120^\circ\) and \(x = 240^\circ\) since cosine is negative in the second and third quadrants.

Key concepts

Cosine Function

The cosine function, denoted as \( \cos \), is a fundamental trigonometric function arising from the ratios in a right-angled triangle or, more broadly, the unit circle. It describes the relationship between the angle \( x \) and the length of the adjacent side divided by the length of the hypotenuse in a right triangle. On the unit circle, \( \cos x \) represents the x-coordinate of a point that lies on the circle, which has a radius of 1. In the equation we're tackling, \( \cos x \) is treated as the variable \( u \), leading to a quadratic equation that we can solve using algebraic techniques.

Unit Circle

The unit circle is a powerful tool in trigonometry, defined as a circle with a radius of one unit centered at the origin of a coordinate system. It allows us to define the sine and cosine functions geometrically and is fundamental to the study of periodic functions. Angles on the unit circle are traditionally measured from the positive x-axis (0 radians or degrees) in a counterclockwise direction for positive angles and clockwise for negative angles.

Each point \( (x, y) \) on the unit circle corresponds to an angle \( \theta \) where \( x = \cos(\theta) \) and \( y = \sin(\theta) \). This concept was used to find the values of \( x \) where \( \cos x = -\frac{1}{2} \) by looking at the angles for which the points on the unit circle have an x-coordinate of \( -\frac{1}{2} \), leading us to the solutions \( x = 120^\circ \) and \( x = 240^\circ \) for our original exercise problem.