Question

Todd believes that \(180^{\circ}\) and \(270^{\circ}\) are solutions to the equation \(5 \cos ^{2} \theta=-4 \cos \theta .\) Show how you would check to determine whether Todd's solutions are correct.

Short answer

Todd's solution \( 270^{\text{\degree}} \) is correct, but \( 180^{\text{\degree}} \) is incorrect.

Step-by-step solution

Simplify the Equation

Start with the given equation: \[ 5 \, \text{cos}^2 \theta = -4 \, \text{cos} \theta. \]To simplify, move all terms to one side of the equation to set it to zero: \[ 5 \, \text{cos}^2 \theta + 4 \, \text{cos} \theta = 0. \]

Factor Out the Common Term

Factor out the common term, which is \( \text{cos} \theta \): \[ \text{cos} \theta (5 \, \text{cos} \theta + 4) = 0. \]

Set Each Factor to Zero

Set each factor equal to zero and solve for \( \theta \): \[ \text{cos} \theta = 0 \] \[ 5 \, \text{cos} \theta + 4 = 0 \]

Solve \( \text{cos} \theta = 0 \)

Solve the equation \( \text{cos} \theta = 0 \): \[ \text{cos} \theta = 0 \Rightarrow \theta = 90^{\text{\degree}}, 270^{\text{\degree}} \, (\text{within a 0}^\circ \text{ to 360}^\circ \text{ range}). \]

Solve \( 5 \, \text{cos} \theta + 4 = 0 \)

Solve the equation \( 5 \, \text{cos} \theta + 4 = 0 \): \[ 5 \, \text{cos} \theta + 4 = 0 \Rightarrow 5 \, \text{cos} \theta = -4 \Rightarrow \text{cos} \theta = -\frac{4}{5}. \]Since \( \text{cos} \theta = -\frac{4}{5} \), find the actual angles \( \theta \) using the inverse cosine function. These solutions do not provide standard angle measurements within our range and require a calculator.

Verify Todd's Solutions

To verify Todd's solutions, check if \( \theta = 180^{\text{\degree}} \) and \( \theta = 270^{\text{\degree}} \) satisfy the original equation.For \( \theta = 180^{\text{\degree}} \):\[ \text{cos} \theta = \text{cos}(180^{\text{\degree}}) = -1 \]Substitute in the original equation:\[ 5(-1)^2 + 4(-1) = 5 - 4 = 1 eq 0 \]So, \( 180^{\text{\degree}} \) is not a solution.For \( \theta = 270^{\text{\degree}} \):\[ \text{cos} \theta = \text{cos}(270^{\text{\degree}}) = 0 \]Substitute in the original equation:\[ 5(0)^2 + 4(0) = 0 \]So, \( 270^{\text{\degree}} \) is a solution.

Key concepts

Solving Trigonometric Equations

Solving trigonometric equations is essential in various branches of mathematics and physics. To find the solutions, you often need to manipulate and simplify the equation. In this method, you set the equation to zero and factorize it, which allows you to solve for the angle in question. Let's look at our equation: \[ 5 \text{cos}^2 \theta + 4 \text{cos} \theta = 0 \] \ Setting the equation to zero makes it easier to apply algebraic methods such as factoring.

Cosine Function

The cosine function, denoted as \(\cos \theta\), is one of the primary trigonometric functions. It measures the horizontal distance of a point on a unit circle as you rotate around the circle. The function has distinct values depending on the angle \(\theta\). For example:
\[\cos(0^{\circ}) = 1\]
\[\cos(90^{\circ}) = 0\]
\[\cos(180^{\circ}) = -1\]
\[\cos(270^{\circ}) = 0\]
In the given problem, the values of \(\cos \theta\) must satisfy the equation \(5 \, \text{cos}^2 \theta + 4 \, \text{cos} \theta = 0\).

Zero-Product Property

The Zero-Product Property is an essential algebraic principle. It states that if the product of two factors is zero, then at least one of the factors must be zero. This principle can simplify solving equations like our trigonometric one. From the equation:
\[\cos \theta (5 \, \cos \theta + 4) = 0\] We conclude that either:
\[\cos \theta = 0\]
or
\[5 \, \cos \theta + 4 = 0\]
These conditions give us specific angles or values for \(\theta\) that we can then examine further.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. They are useful tools for simplifying and solving trigonometric equations. Common identities include the Pythagorean identities, angle sum and difference identities, as well as double and half-angle formulas. For solving the given problem, factoring and the Zero-Product Property were used without delving into deeper identities.
However, knowing that:
\[ \cos \theta = 0 \rightarrow \theta = 90^{\circ}, 270^{\circ}\] within the standard angle range is crucial. Similarly, understanding that:
\[ \cos \theta = -\frac{4}{5} \rightarrow \theta \approx 126.87^{\circ}, 233.13^{\circ} \] can help verify the solutions in detailed problems.