Solve The Following Equation For All Values Of 0 ≤ Θ \textless 2 Π 0 \leq \theta \ \textless \ 2\pi 0 ≤ Θ \textless 2 Π . 4 Cos ⁡ 2 Θ + 2 Cos ⁡ Θ − 2 = 0 4 \cos^2 \theta + 2 \cos \theta - 2 = 0 4 Cos 2 Θ + 2 Cos Θ − 2 = 0 Provide Your Answer Below: Θ = □ \theta = \square Θ = □

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific trigonometric equation involving cosine, and provide a step-by-step guide on how to approach it.

The Equation

The given equation is:

4cos⁡2θ+2cos⁡θ−2=04 \cos^2 \theta + 2 \cos \theta - 2 = 0

where 0≤θ<2π0 \leq \theta < 2\pi. Our goal is to solve for all values of θ\theta that satisfy this equation.

Step 1: Factor the Quadratic Expression

The first step in solving this equation is to factor the quadratic expression on the left-hand side. We can start by factoring out the greatest common factor (GCF), which is 2:

2(2cos⁡2θ+cos⁡θ−1)=02(2 \cos^2 \theta + \cos \theta - 1) = 0

Step 2: Use the Quadratic Formula

The quadratic expression inside the parentheses can be factored using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=2a = 2, b=1b = 1, and c=−1c = -1. Plugging these values into the formula, we get:

cos⁡θ=−1±1+84\cos \theta = \frac{-1 \pm \sqrt{1 + 8}}{4}

Step 3: Simplify the Expression

Simplifying the expression inside the square root, we get:

cos⁡θ=−1±94\cos \theta = \frac{-1 \pm \sqrt{9}}{4}

cos⁡θ=−1±34\cos \theta = \frac{-1 \pm 3}{4}

Step 4: Solve for cos⁡θ\cos \theta

Now we have two possible values for cos⁡θ\cos \theta:

cos⁡θ=−1+34=24=12\cos \theta = \frac{-1 + 3}{4} = \frac{2}{4} = \frac{1}{2}

cos⁡θ=−1−34=−44=−1\cos \theta = \frac{-1 - 3}{4} = \frac{-4}{4} = -1

Step 5: Find the Values of θ\theta

Now that we have the values of cos⁡θ\cos \theta, we can find the corresponding values of θ\theta using the inverse cosine function:

θ=cos⁡−1(12)\theta = \cos^{-1} \left( \frac{1}{2} \right)

θ=cos⁡−1(−1)\theta = \cos^{-1} (-1)

Step 6: Simplify the Expressions

Using the unit circle or a calculator, we can find the values of θ\theta:

θ=π3\theta = \frac{\pi}{3}

θ=π\theta = \pi

Conclusion

In this article, we solved a trigonometric equation involving cosine using a step-by-step approach. We factored the quadratic expression, used the quadratic formula, simplified the expression, solved for cos⁡θ\cos \theta, and finally found the values of θ\theta. The solutions to the equation are θ=π3\theta = \frac{\pi}{3} and θ=π\theta = \pi.

Final Answer

The final answer is:

θ=π3,π\theta = \boxed{\frac{\pi}{3}, \pi}

Discussion

This equation is a quadratic equation in terms of cos⁡θ\cos \theta. We can use the quadratic formula to solve for cos⁡θ\cos \theta, and then find the corresponding values of θ\theta using the inverse cosine function. This approach can be applied to other trigonometric equations involving sine and tangent as well.

Tips and Variations

  • To solve this equation, we can also use the identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1 to rewrite the equation in terms of sin⁡θ\sin \theta.
  • We can also use the double-angle formula for cosine to rewrite the equation in terms of cos⁡2θ\cos 2\theta.
  • To find the values of θ\theta that satisfy the equation, we can use the unit circle or a calculator to find the inverse cosine of the values of cos⁡θ\cos \theta.

Related Topics

  • Trigonometric identities
  • Quadratic equations
  • Inverse trigonometric functions
  • Unit circle

Glossary

  • Trigonometric equation: An equation that involves trigonometric functions such as sine, cosine, and tangent.
  • Quadratic expression: An expression that can be written in the form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants.
  • Inverse cosine function: A function that returns the angle whose cosine is a given value.
  • Unit circle: A circle with a radius of 1 that is centered at the origin of a coordinate plane.

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Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, and tangent. These equations can be used to model real-world problems and can be solved using various techniques.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use various techniques such as factoring, using the quadratic formula, and applying trigonometric identities. You can also use inverse trigonometric functions to find the values of the trigonometric functions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to solve a trigonometric equation?

A: To use the quadratic formula to solve a trigonometric equation, you can first rewrite the equation in the form of a quadratic equation. Then, you can apply the quadratic formula to find the values of the trigonometric function.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that are used to describe the relationships between the angles and side lengths of triangles. The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I use the unit circle to solve a trigonometric equation?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. You can use the unit circle to find the values of the trigonometric functions by drawing a line from the origin to a point on the circle that corresponds to the angle you are interested in.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Q: How do I use inverse trigonometric functions to solve a trigonometric equation?

A: To use inverse trigonometric functions to solve a trigonometric equation, you can first rewrite the equation in the form of a trigonometric function. Then, you can apply the inverse trigonometric function to find the value of the angle.

Q: What are some common applications of trigonometric equations?

A: Some common applications of trigonometric equations include:

  • Modeling the motion of objects in physics and engineering
  • Describing the relationships between the angles and side lengths of triangles in geometry
  • Solving problems in navigation and surveying
  • Modeling population growth and decay in biology and economics

Q: How do I choose the correct trigonometric function to use in a problem?

A: To choose the correct trigonometric function to use in a problem, you should consider the relationships between the angles and side lengths of the triangle. You should also consider the units of measurement and the context of the problem.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

  • Failing to simplify the equation before solving it
  • Using the wrong trigonometric function or identity
  • Failing to check the solutions for extraneous solutions
  • Failing to consider the domain and range of the trigonometric function

Q: How do I check my solutions for extraneous solutions?

A: To check your solutions for extraneous solutions, you should plug the solutions back into the original equation and check if they are true. You should also check if the solutions are within the domain and range of the trigonometric function.

Q: What are some common resources for learning about trigonometric equations?

A: Some common resources for learning about trigonometric equations include:

  • Textbooks and online resources such as Khan Academy and MIT OpenCourseWare
  • Online tutorials and video lectures
  • Practice problems and worksheets
  • Online communities and forums for math enthusiasts

Q: How do I practice solving trigonometric equations?

A: To practice solving trigonometric equations, you should start by working through practice problems and worksheets. You should also try to solve problems on your own before checking the solutions. You can also join online communities and forums for math enthusiasts to get help and feedback on your work.