Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ 5 \cos ^{2} \alpha+3 \cos \alpha-2=0 $$

Step-by-Step Solution

Answer

The approximate solutions are \(70^\circ\) and \(290^\circ\).

Step 1: Rewrite the Equation

The equation is given as \(5 \cos^2 \alpha + 3 \cos \alpha - 2 = 0\). This is a quadratic equation in terms of \(\cos \alpha\). Let's rewrite it as \(5x^2 + 3x - 2 = 0\), where \(x = \cos \alpha\).

Step 2: Solve the Quadratic Equation

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for \(x\) where \(a = 5\), \(b = 3\), and \(c = -2\). Calculate the discriminant: \(b^2 - 4ac = 3^2 - 4(5)(-2) = 9 + 40 = 49\).

Step 3: Calculate the Roots

Substitute the discriminant into the quadratic formula: \[x = \frac{-3 \pm \sqrt{49}}{10}\]This gives \(x = \frac{-3 + 7}{10} = 0.4\) and \(x = \frac{-3 - 7}{10} = -1\). Only \(0.4\) is valid since \(\cos \alpha\) ranges from -1 to 1.

Step 4: Find Angles for Valid Cosine Values

For \(\cos \alpha = 0.4\), find \(\alpha\) such that \(\alpha = \cos^{-1}(0.4)\) giving approximately \(66.4^\circ\). Considering the range \([0^\circ, 360^\circ)\), the second angle is \(360^\circ - 66.4^\circ = 293.6^\circ\).

Step 5: Round to Nearest 10 Degrees

Round \(66.4^\circ\) to \(70^\circ\) and \(293.6^\circ\) to \(290^\circ\) to the nearest 10 degrees.

Key Concepts

Quadratic Equations

To tackle the problem, we first recognize it as a quadratic equation, which is a polynomial equation of the second degree. The general form is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our context, \( x \) is represented by \( \cos \alpha \). A quadratic equation can have up to two solutions or roots, which can be found using various methods like factoring, completing the square, or using the quadratic formula, which is a universal solution method.
The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) allows us to find these roots by substituting the appropriate values for \( a \), \( b \), and \( c \). The expression under the square root, \( b^2 - 4ac \), is known as the discriminant and determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it's zero, there is one real root.
• If it's negative, the roots are complex.

In our exercise, the discriminant was positive, granting us two real roots to work with.

Cosine Function

The cosine function, denoted as \( \cos \), is a fundamental function in trigonometry that describes the relationship between the angles and sides of a right triangle. It is one of the primary trigonometric functions and is periodic with a period of \( 360^\circ \) or \( 2\pi \) radians.
Cosine values range between -1 and 1, signifying the adjacent side over the hypotenuse in a right triangle. For a given angle \( \alpha \), \( \cos \alpha \) is the x-coordinate of the point on the unit circle at angle \( \alpha \) from the positive x-axis.
This trigonometric function is crucial due to its symmetrical properties:

• Zeros at \( 90^\circ + 180^\circ k \), where \( k \) is an integer.
• Even symmetry, meaning \( \cos(-\alpha) = \cos(\alpha) \).

For the exercise in question, we solved \( \cos \alpha = 0.4 \) to find the angle \( \alpha \) that relates to this specific cosine value.

Angle Approximation

Angle approximation is an essential mathematical technique, especially when dealing with trigonometric solutions. In this exercise, the computed angle from \( \cos^{-1}(0.4) \) resulted in \( 66.4^\circ \). However, working with exact decimal angles is not always practical.
The main goal of angle approximation is to estimate angles to values that are easier to work with, without significant loss of precision in everyday applications. In trigonometry, especially, precise angles are often rounded to the nearest whole number or a specific degree setting for simplicity.
This helps especially in:

• Creating easier computations in physics and engineering tasks.
• Enhancing readability and understanding of results.

Through approximation, the second angle in our solution was deduced using symmetry at approximately \( 293.6^\circ \). Though an exact angle down to decimals might be crucial for high precision measurements, approximations offer a more feasible approach for general understanding.

Degree Rounding

When dealing with angles in practical scenarios, rounding to the nearest specified degree significantly simplifies interpretation and application. Degree rounding is particularly beneficial in fields such as navigation, physics, and computer graphics, where exact decimals may not be as practical.
In our exercise, understanding degree rounding requires us to look at the nearest tens. For instance, the exact angle \( 66.4^\circ \) is rounded to \( 70^\circ \) because it is closer to 70 than to 60. Similarly, \( 293.6^\circ \) rounds down to \( 290^\circ \), as it is closer to 290.
Key steps in degree rounding include:

• Identifying the nearest whole number.
• Determining the degree of precision required (e.g., nearest 10').
• Applying conventional rounding rules (e.g., values of 0.5 and above round up).

This approach helps provide a clear and concise representation of angles, allowing users to focus more on the application rather than the complexities of exact values.