Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\cos x=-1$$

Step-by-Step Solution

Answer

The solution to the given trigonometric equation \(\cos x = -1\) in the range from \(0^{\circ}\) to \(360^{\circ}\) is \(x=180^{\circ}\).

Step 1: Understand the Cosine Function

The first step in solving this problem is understanding the cosine function. Cosine function gives the x-coordinate of a point on the unit circle determined by the angle measured in degrees or radian from the positive x-axis.

Step 2: Identify the Range

The next step is to observe the given range for the solution, that is from \(0^{\circ}\) to \(360^{\circ}\). The cosine function repeats every \(360^{\circ}\). So, it's enough to check only in this range.

Step 3: Solve the Equation

The next step is to actually solve the given equation, \(\cos x = -1\). The cosine function is -1 at \(x=180^{\circ}\) within the given range.

Key Concepts

The Cosine Function

The cosine function is one of the fundamental trigonometric functions and is closely tied to the shape of a right triangle and the unit circle.

• In a right triangle: The cosine of an angle is defined as the ratio of the length of the adjacent side over the hypotenuse.
• As a function: \(\cos(\theta) = \text{adjacent} / \text{hypotenuse}\)
• Graphically: The cosine function appears as a wave that oscillates between -1 and 1, repeating every 360 degrees (or \(2\pi\) radians).
• On the unit circle: The cosine function will give you the x-coordinate of a point on the circumference for a given angle \(\theta\).

Understanding where the cosine function takes on values such as \(0\), \(1\), or \(-1\) is crucial for solving equations like \(\cos x = -1\). In trigonometry, identifying these key points helps solve equations efficiently.

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit centered at the origin \((0, 0)\) in the coordinate plane.

• Coordinates and angles: On the unit circle, each point represents the endpoint of an angle measured from the positive x-axis, moving counterclockwise. The x-coordinate of the point is the value of \(\cos \theta\), and the y-coordinate is \(\sin \theta\).
• Full circle: A complete revolution around the unit circle is \(360^{\circ}\) (or \(2\pi\) radians). With this, we can effectively understand the periodicity of trigonometric functions like sine and cosine.
• Key positions: Important angles like \(0^{\circ}\) \((1, 0)\), \(90^{\circ}\) \((0, 1)\), \(180^{\circ}\) \((-1, 0)\), and \(270^{\circ}\) \((0, -1)\) define cosine values (-1 or 1) that help solve equational trick points.

In the problem \(\cos x = -1\), the solution \(180^{\circ}\) is identified because the x-coordinate at \(180^{\circ}\) gives \(-1\), reinforcing our understanding of angles and coordinates on the unit circle.

Degrees and Radians

Degrees and radians are two units that measure angles.

• Degrees: A circle is divided into 360 degrees, where \(0^{\circ}\) is at the positive x-axis, and the angle increases counterclockwise. This method is highly intuitive and widely used in many fields including navigation and mapping.
• Radians: This measure is based on the radius of the circle. One full circle in radians is \(2\pi\). For instance, \(180^{\circ}\) converts to \(\pi\) radians.
• Conversion: To convert degrees to radians, use the formula \(\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)\). Conversely, to convert from radians to degrees, use the formula \(\text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right)\).

In trigonometric equations, being comfortable with both units is important. Converting between them allows you to apply concepts interchangeably, ensuring full comprehension regardless of the form in which equations appear. When solving \(\cos x = -1\) in the context of \(0^{\circ}\) to \(360^{\circ}\), it's important to remember that \(180^{\circ}\) is equivalent to \(\pi\), which would also solve the equation in radians.