Solve For Θ \theta Θ : Cos ⁡ 2 Θ + 4 Cos ⁡ Θ = − 3 \cos 2\theta + 4\cos\theta = -3 Cos 2 Θ + 4 Cos Θ = − 3

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Introduction

Solving trigonometric equations can be a challenging task, especially when they involve multiple terms and complex expressions. In this article, we will focus on solving the equation $\cos 2\theta + 4\cos\theta = -3$ for the variable $\theta$. This equation involves the cosine function and its double angle identity, making it a great example of how to apply trigonometric identities to solve equations.

Understanding the Equation

The given equation is $\cos 2\theta + 4\cos\theta = -3$. To solve this equation, we need to isolate the variable $\theta$. However, the equation involves the cosine function and its double angle identity, which makes it difficult to solve directly. We need to use trigonometric identities to simplify the equation and make it easier to solve.

Using Trigonometric Identities

One of the most useful trigonometric identities is the double angle identity for cosine, which states that $\cos 2\theta = 2\cos^2 \theta - 1$. We can use this identity to rewrite the equation $\cos 2\theta + 4\cos\theta = -3$ as $2\cos^2 \theta - 1 + 4\cos\theta = -3$.

Simplifying the Equation

Now that we have rewritten the equation using the double angle identity, we can simplify it further. We can start by combining like terms: $2\cos^2 \theta + 4\cos\theta - 2 = -3$. Next, we can add 2 to both sides of the equation to get $2\cos^2 \theta + 4\cos\theta = -1$.

Rearranging the Equation

To make it easier to solve the equation, we can rearrange it to get $2\cos^2 \theta + 4\cos\theta + 1 = 0$. This is a quadratic equation in terms of $\cos\theta$, and we can solve it using the quadratic formula.

Solving the Quadratic Equation

The quadratic formula states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our case, $a = 2$, $b = 4$, and $c = 1$. Plugging these values into the quadratic formula, we get $\cos\theta = \frac{-4 \pm \sqrt{4^2 - 4(2)(1)}}{2(2)}$.

Simplifying the Quadratic Formula

Simplifying the quadratic formula, we get $\cos\theta = \frac{-4 \pm \sqrt{16 - 8}}{4}$. This simplifies to $\cos\theta = \frac{-4 \pm \sqrt{8}}{4}$.

Simplifying the Square Root

The square root of 8 can be simplified as $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$. Therefore, the quadratic formula simplifies to $\cos\theta = \frac{-4 \pm 2\sqrt{2}}{4}$.

Simplifying the Expression

We can simplify the expression further by dividing both the numerator and the denominator by 2. This gives us $\cos\theta = \frac{-2 \pm \sqrt{2}}{2}$.

Finding the Values of $\theta$

Now that we have found the values of $\cos\theta$, we can use the inverse cosine function to find the values of $\theta$. The inverse cosine function is denoted by $\cos^{-1}$ and is defined as the angle whose cosine is a given value.

Using the Inverse Cosine Function

Using the inverse cosine function, we can find the values of $\theta$ as follows:

$\theta = \cos^{-1} \left( \frac{-2 + \sqrt{2}}{2} \right)$ and $\theta = \cos^{-1} \left( \frac{-2 - \sqrt{2}}{2} \right)$.

Evaluating the Inverse Cosine Function

Evaluating the inverse cosine function, we get:

$\theta = \cos^{-1} \left( \frac{-2 + \sqrt{2}}{2} \right) = \cos^{-1} \left( -\frac{1 + \sqrt{2}}{2} \right)$ and $\theta = \cos^{-1} \left( \frac{-2 - \sqrt{2}}{2} \right) = \cos^{-1} \left( \frac{1 + \sqrt{2}}{2} \right)$.

Finding the Values of $\theta$ in Radians

Using a calculator, we can find the values of $\theta$ in radians as follows:

$\theta = \cos^{-1} \left( -\frac{1 + \sqrt{2}}{2} \right) \approx 2.53$ radians and $\theta = \cos^{-1} \left( \frac{1 + \sqrt{2}}{2} \right) \approx 4.71$ radians.

Finding the Values of $\theta$ in Degrees

Using a calculator, we can find the values of $\theta$ in degrees as follows:

$\theta = \cos^{-1} \left( -\frac{1 + \sqrt{2}}{2} \right) \approx 145.25$ degrees and $\theta = \cos^{-1} \left( \frac{1 + \sqrt{2}}{2} \right) \approx 27.74$ degrees.

Conclusion

In this article, we have solved the equation $\cos 2\theta + 4\cos\theta = -3$ for the variable $\theta$. We used the double angle identity for cosine to rewrite the equation and then simplified it to a quadratic equation in terms of $\cos\theta$. We solved the quadratic equation using the quadratic formula and then used the inverse cosine function to find the values of $\theta$. We evaluated the inverse cosine function to find the values of $\theta$ in radians and degrees.

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Introduction

In our previous article, we solved the equation $\cos 2\theta + 4\cos\theta = -3$ for the variable $\theta$. We used the double angle identity for cosine to rewrite the equation and then simplified it to a quadratic equation in terms of $\cos\theta$. We solved the quadratic equation using the quadratic formula and then used the inverse cosine function to find the values of $\theta$. In this article, we will answer some common questions related to solving this equation.

Q: What is the double angle identity for cosine?

A: The double angle identity for cosine is $\cos 2\theta = 2\cos^2 \theta - 1$. This identity is used to rewrite the equation $\cos 2\theta + 4\cos\theta = -3$ in terms of $\cos\theta$.

Q: How do I simplify the equation $\cos 2\theta + 4\cos\theta = -3$?

A: To simplify the equation, we can use the double angle identity for cosine to rewrite the equation as $2\cos^2 \theta - 1 + 4\cos\theta = -3$. We can then combine like terms and rearrange the equation to get $2\cos^2 \theta + 4\cos\theta + 1 = 0$.

Q: How do I solve the quadratic equation $2\cos^2 \theta + 4\cos\theta + 1 = 0$?

A: We can solve the quadratic equation using the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our case, $a = 2$, $b = 4$, and $c = 1$. Plugging these values into the quadratic formula, we get $\cos\theta = \frac{-4 \pm \sqrt{4^2 - 4(2)(1)}}{2(2)}$.

Q: What is the inverse cosine function?

A: The inverse cosine function is denoted by $\cos^{-1}$ and is defined as the angle whose cosine is a given value. We can use the inverse cosine function to find the values of $\theta$ from the solutions of the quadratic equation.

Q: How do I find the values of $\theta$ in radians and degrees?

A: We can use a calculator to find the values of $\theta$ in radians and degrees. For example, if we have a solution of the form $\cos\theta = \frac{-2 + \sqrt{2}}{2}$, we can use the inverse cosine function to find the value of $\theta$ in radians and degrees.

Q: What are some common mistakes to avoid when solving the equation $\cos 2\theta + 4\cos\theta = -3$?

A: Some common mistakes to avoid when solving the equation include:

• Not using the double angle identity for cosine to rewrite the equation
• Not combining like terms and rearranging the equation correctly
• Not using the quadratic formula to solve the quadratic equation
• Not using the inverse cosine function to find the values of $\theta$
• Not checking the solutions for extraneous solutions

Q: How can I check if a solution is extraneous?

A: To check if a solution is extraneous, we can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some real-world applications of solving the equation $\cos 2\theta + 4\cos\theta = -3$?

A: Solving the equation $\cos 2\theta + 4\cos\theta = -3$ has many real-world applications, including:

• Modeling the motion of a pendulum
• Modeling the motion of a spring
• Modeling the behavior of a electrical circuit
• Modeling the behavior of a mechanical system

Conclusion

In this article, we have answered some common questions related to solving the equation $\cos 2\theta + 4\cos\theta = -3$. We have discussed the double angle identity for cosine, how to simplify the equation, how to solve the quadratic equation, and how to find the values of $\theta$ in radians and degrees. We have also discussed some common mistakes to avoid and how to check if a solution is extraneous. Finally, we have discussed some real-world applications of solving the equation.