Problem 38 - Trigonometry

Question

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$\cos 2 \theta+3 \cos \theta-2=0$$

Step-by-Step Solution

Answer

The solutions are \( \theta \approx 46.5^{\circ} \) and \( \theta \approx 313.5^{\circ} \).

Step 1: Rewrite the Equation

The given equation is \( \cos 2\theta + 3\cos \theta - 2 = 0 \). Use the double angle identity \( \cos 2\theta = 2\cos^2\theta - 1 \) to rewrite the equation. Substitute it into the equation to get \( 2\cos^2\theta - 1 + 3\cos\theta - 2 = 0 \).

Step 2: Simplify the Equation

Simplify the equation: \( 2\cos^2\theta + 3\cos\theta - 3 = 0 \).

Step 3: Substitute Variable

Let \( x = \cos\theta \). Substitute \( x \) into the equation to get a quadratic equation: \( 2x^2 + 3x - 3 = 0 \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = 3 \), \( c = -3 \). Calculate the discriminant: \( b^2 - 4ac = 3^2 - 4(2)(-3) = 9 + 24 = 33 \). Then, solve for \( x \): \( x = \frac{-3 \pm \sqrt{33}}{4} \).

Step 5: Find the Values of \(\cos\theta\)

The solutions to the quadratic equation are \( x = \frac{-3 + \sqrt{33}}{4} \) and \( x = \frac{-3 - \sqrt{33}}{4} \). Calculate these values: \( x_1 \approx 0.686 \) and \( x_2 \approx -2.186 \). Only \( x_1 \) is within the valid range of cosine values \([ -1, 1 ]\).

Step 6: Find \(\theta\) for \(\cos\theta = x_1\)

Use the inverse cosine function to find \( \theta \): \( \theta = \cos^{-1}(0.686) \). Calculate \( \theta \approx 46.5^{\circ} \).

Step 7: Determine All Solutions in Given Interval

Since cosine is positive in both the first and fourth quadrants, \( \theta \approx 46.5^{\circ} \) and the second solution in the fourth quadrant is \( \theta \approx 360^{\circ} - 46.5^{\circ} = 313.5^{\circ} \). Therefore, the solutions in the interval \(0^{\circ} \leq \theta < 360^{\circ} \) are \( \theta \approx 46.5^{\circ} \) and \( \theta \approx 313.5^{\circ} \).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra. They are mathematical expressions of the form \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic equations appear frequently in problems involving curves, optimization, and are even integral in solving trigonometric equations. To find the roots (solutions) of a quadratic equation, one of the most common methods used is the quadratic formula, given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The expression under the square root, \( b^2 - 4ac \), is known as the discriminant.
- If the discriminant is positive, there are two distinct real roots.- If it is zero, there is exactly one real root (or a repeated root).- If the discriminant is negative, there are no real roots, but rather two complex roots.In our exercise, substituting \( x = \cos\theta \) allowed us to express the trigonometric equation in a familiar quadratic form, making it easier to solve using the quadratic formula. The transformation from a trigonometric equation to a quadratic one illustrates the power of substitution in problem-solving.

Cosine Function

The cosine function, \( \cos\theta \), is one of the primary trigonometric functions. It relates the angle \( \theta \) in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. Cosine is fundamental in modeling periodic phenomena such as sound waves, light waves, and alternating current.

• Range and Domain: The range of the cosine function is \[-1 \leq \cos\theta \leq 1\], while its domain is all real numbers because angles can be positive, negative, or zero.
• Periodicity: The cosine function is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians, meaning it repeats its values every \(360^{\circ}\).
• Symmetry: Cosine is an even function, which means \(\cos(-\theta) = \cos\theta\), giving it symmetry about the y-axis.

The solving process for our equation required a deep understanding of the cosine function's properties. Especially crucial was the knowledge of its range, which ensured that we only focused on solutions for \(\cos\theta\) that fell between -1 and 1.

Angle Identities

Angle identities are powerful tools in trigonometry. They allow the simplification of expressions and the solving of equations by relating various trigonometric functions. In the context of our exercise, we used the double angle identity for cosine:\[ \cos 2\theta = 2\cos^2\theta - 1 \]This identity lets us express \(\cos 2\theta \) in terms of \( \cos\theta \), simplifying the original equation.

• Other Common Angle Identities: There are numerous angle identities, including the Pythagorean identities, sum and difference formulas, and half-angle identities.
• Application: These identities are not just theoretical but practical. They are crucial when transforming complex trigonometric expressions into simpler forms that are easier to evaluate or solve.

Applying the double angle identity in the step-by-step solution allowed us to restate our problem in a manageable form, reducing it to an equation about \(\cos\theta\). Mastery of such identities therefore gives you an important edge when tackling a broad range of trigonometric problems.