Problem 65 - Precalculus with Calculus Previews

Question

In Problems \(65-70\), using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ 20 \cos ^{2} x+\cos x-1=0,[0, \pi] $$

Step-by-Step Solution

Answer

The solutions are approximately \(x = 1.37\) and \(x = 1.82\) radians.

Step 1: Use a substitution

To simplify the equation, let's substitute \(y = \cos x\). Then the equation becomes \(20y^2 + y - 1 = 0\).

Step 2: Use the Quadratic Formula

The equation \(20y^2 + y - 1 = 0\) is a quadratic equation in terms of \(y\). We can use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 20\), \(b = 1\), and \(c = -1\).

Step 3: Calculate the Discriminant

First, compute the discriminant: \(b^2 - 4ac = 1^2 - 4\cdot20\cdot(-1) = 1 + 80 = 81\).

Step 4: Solve for \(y\) using the Quadratic Formula

Since the discriminant is 81, we proceed: \(y = \frac{-1 \pm \sqrt{81}}{2 \cdot 20}\) which simplifies to \(y = \frac{-1 \pm 9}{40}\). The solutions are \(y = \frac{8}{40} = \frac{1}{5}\) and \(y = \frac{-10}{40} = -\frac{1}{4}\).

Step 5: Calculate \(x\) for \(y = \frac{1}{5}\)

Using the inverse cosine function, \(x = \cos^{-1}\left(\frac{1}{5}\right)\). Calculating this gives approximately \(x = 1.37\) radians.

Step 6: Calculate \(x\) for \(y = -\frac{1}{4}\)

Similarly, use \(x = \cos^{-1}\left(-\frac{1}{4}\right)\). Calculating this gives approximately \(x = 1.82\) radians.

Step 7: Verify Solutions in Domain

Check both solutions \(x = 1.37\) and \(x = 1.82\) radians to ensure they lie within the interval \([0, \pi]\). Both values are within this interval.

Key Concepts

Quadratic Formula

The quadratic formula is a mathematical tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula is essential because it provides a straightforward way to find the solutions or roots of such equations. Let's break it down further:

• Formula Structure: The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where:
  • \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^2 + bx + c = 0\).
• Discriminant: The part under the square root, \(b^2 - 4ac\), is called the discriminant.
  • If the discriminant is positive, there are two real solutions.
  • If it is zero, there is one real solution.
  • If it is negative, the solutions are complex numbers.

By substituting the coefficients \(a = 20\), \(b = 1\), and \(c = -1\) into the formula, we can find the roots of the equation \(20y^2 + y - 1 = 0\). In the given problem, the discriminant was \(81\), indicating two distinct real solutions.

Cosine Function

The cosine function is one of the primary trigonometric functions, often abbreviated as \(\cos\). It's fundamental in understanding wave patterns, circular motion, and oscillations.

• Definition: For an angle \(x\), \(\cos x\) is the x-coordinate of the point on the unit circle that corresponds to the angle \(x\).
• Inverse Cosine: The inverse function, denoted \(\cos^{-1} x\) or \(\text{arccos}(x)\), finds an angle whose cosine is \(x\). Its range is \([0, \pi]\), making it useful for solving equations within this interval.

In the problem, we used the cosine function to bridge the solutions from the quadratic equation to solutions for \(x\), since \(y = \cos x\). For instance, after solving for \(y = \frac{1}{5}\) and \(y = -\frac{1}{4}\), we found \(x\) using the inverse cosine values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables for which both sides of the identity are defined. They are valuable in simplifying complex trigonometric expressions and solving equations.

• Basic Identities: Some common identities include
  • \( \cos^2 x + \sin^2 x = 1 \)
  • \( \sec x = \frac{1}{\cos x} \)
• Applications: They are often used to transform a trigonometric equation into a more manageable form, such as converting all terms into sines and cosines or reducing powers.

In solving the exercise, understanding basic trigonometric identities would allow us to confirm the correctness of the given cosine-related solutions and verify them within specified intervals.